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<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp; <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">h</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b2">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> &amp; <font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b4">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> ) &amp; ( for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">e</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> &amp; not <font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">h</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp; not <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) ) )</span><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  &amp; <font color="Maroon" title="c2">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> &amp; not <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
 <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">g1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:27"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c3">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c3">g1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green" title="E28">A5</font></i>: 
( <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="jordan7.html#T18" target="_self" title="JORDAN7:th.18">Th18</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E29">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green" title="E30">A7</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E23"><i><font color="Green" title="E20">Lm5</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green" title="E31">A8</font></i>: 
(  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">h1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E11:27"/><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c4">h1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">Q</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="uniform1.html#T15" title="UNIFORM1:th.15">UNIFORM1:15</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c4">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="jordan7.html#T17" target="_self" title="JORDAN7:th.17">Th17</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green" title="E34">A11</font></i>: 
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c5">g2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c5">g2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>and </b><br/><a NAME="E16:27"/><i><font color="Green" title="E36">A13</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>and </b><br/><a NAME="E17:27"/><i><font color="Green" title="E37">A14</font></i>: 
( <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="jordan7.html#T18" target="_self" title="JORDAN7:th.18">Th18</a></i>;<br/>
<a NAME="E18:27"/><i><font color="Green" title="E38">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">h2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E20:27"/><i><font color="Green" title="E39">A16</font></i>: 
( <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c6">h2</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b3">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">Q</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b3">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="uniform1.html#T15" title="UNIFORM1:th.15">UNIFORM1:15</a></i>;<br/>
<a NAME="E21:27"/><i><font color="Green" title="E40">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E22"><i><font color="Green" title="E19">Lm4</font></i></a></i>;<br/>
<a NAME="E22:27"/><i><font color="Green" title="E41">A18</font></i>: 
<font color="Maroon" title="c6">h2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="jordan7.html#T17" target="_self" title="JORDAN7:th.17">Th17</a></i>;<br/>
<a NAME="E23:27"/>
<font color="Maroon" title="c4">h1</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">h11</font> = <font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E25:27"/><i><font color="Green" title="E42">A19</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:27"/><b>then </b><i><font color="Green" title="E43">A20</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">h11</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E27:27"/><b>then </b><i><font color="Green" title="E44">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E28:27"/>
<font color="Maroon" title="c6">h2</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">h21</font> = <font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T36" title="FINSEQ_2:th.36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E30:27"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E31:27"/><b>then </b><i><font color="Green" title="E45">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E32:27"/><b>then </b><i><font color="Green" title="E46">A23</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E33:27"/><i><font color="Green" title="E47">A24</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:27"/><b>then </b><i><font color="Green" title="E48">A25</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E35:27"/><i><font color="Green" title="E49">A26</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E36:27"/><i><font color="Green" title="E50">A27</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:27"/><b>then </b><i><font color="Green" title="E51">A28</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E38:27"/><i><font color="Green" title="E52">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E39:27"/><i><font color="Green" title="E53">A30</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E40:27"/><i><font color="Green" title="E54">A31</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E41:27"/><i><font color="Green" title="E55">A32</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E23"><i><font color="Green" title="E20">Lm5</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E42:27"/><i><font color="Green" title="E56">A33</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E43:27"/><b>then </b><i><font color="Green" title="E57">A34</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E23"><i><font color="Green" title="E20">Lm5</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E44:27"/><i><font color="Green" title="E58">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E45:27"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green" title="E56">A33</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E46:27"/><b>then </b><i><font color="Green" title="E59">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<a NAME="E47:27"/><b>then </b><i><font color="Green" title="E60">A37</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">h0</font> = <font color="Maroon" title="c7">h11</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E49:27"/><i><font color="Green" title="E61">A38</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green" title="E57">A34</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">h0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<b>thus </b><a NAME="E50:27"/><i><font color="Green" title="E62">A39</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green" title="E57">A34</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">h0</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<a NAME="E51:27"/><i><font color="Green" title="E63">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E52:27"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:27"/><b>then </b><i><font color="Green" title="E64">A41</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E54:27"/><b>then </b><i><font color="Green" title="E65">A42</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E55:27"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E56:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E57:27"/><b>then </b><i><font color="Green" title="E66">A43</font></i>: 
( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E58:27"/>
3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E59:27"/><b>then </b><i><font color="Green" title="E67">A44</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E60:27"/><b>then </b><i><font color="Green" title="E68">A45</font></i>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<a NAME="E61:27"/><b>then </b><i><font color="Green" title="E69">A46</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E62:27"/><i><font color="Green" title="E70">A47</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E63:27"/><b>then </b><i><font color="Green" title="E71">A48</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E64:27"/><i><font color="Green" title="E72">A49</font></i>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E65:27"/><b>then </b><i><font color="Green" title="E73">A50</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E66:27"/><b>then </b><i><font color="Green" title="E74">A51</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E67:27"/><b>then </b><i><font color="Green" title="E75">A52</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E68:27"/><b>then </b><i><font color="Green" title="E76">A53</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E69:27"/><b>then </b><i><font color="Green" title="E77">A54</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E70:27"/><b>then </b><i><font color="Green" title="E78">A55</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E71:27"/><i><font color="Green" title="E79">A56</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E63:27"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E72:27"/><i><font color="Green" title="E80">A57</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E73:27"/><i><font color="Green" title="E81">A58</font></i>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E74:27"/><b>then </b><i><font color="Green" title="E82">A59</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E75:27"/><i><font color="Green" title="E83">A60</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E76:27"/><b>then </b><i><font color="Green" title="E84">A61</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E77:27"/><i><font color="Green" title="E85">A62</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E78:27"/><i><font color="Green" title="E86">A63</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green" title="E76">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E79:27"/><b>then </b><i><font color="Green" title="E87">A64</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E80:27"/><i><font color="Green" title="E88">A65</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E81:27"/><i><font color="Green" title="E89">A66</font></i>: 
<font color="Maroon" title="c7">h11</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>
<a NAME="E82:27"/><i><font color="Green" title="E90">A67</font></i>: 
<font color="Maroon" title="c8">h21</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>
<b>thus </b><a NAME="E83:27"/><i><font color="Green" title="E91">A68</font></i>: 
<font color="Maroon" title="c9">h0</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<font color="Maroon" title="c9">h0</font>) or  not <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relat_1.html#K1" title="RELAT_1:func.1">K1</a>(<font color="Maroon" title="c9">h0</font>) or  not <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E92">A69</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c12">nx</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">ny</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:27_1"/><i><font color="Green" title="E93">A70</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font> &amp; <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:27_1"/><i><font color="Green" title="E94">A71</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> &amp; <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
( <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/>
<font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b><i><font color="Green" title="E95">A72</font></i>: 
<font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_1"><i><font color="Green" title="E93">A70</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_1"/><b>then </b><i><font color="Green" title="E96">A73</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><i><font color="Green" title="E97">A74</font></i>: <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> = 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font>
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">nx</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:27_1_1_1"/><b>then </b><i><font color="Green" title="E98">A75</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1_1_1"><i><font color="Green" title="E96">A73</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_2"/>
( <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_1"/>
<font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E99">A76</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E100">A77</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> = 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font>
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_1_2_1"><i><font color="Green" title="E99">A76</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">ny</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E2:27_1_1_1_2_1"><i><font color="Green" title="E99">A76</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><a NAME="E5:27_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1_1_1"><i><font color="Green" title="E96">A73</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_1"><i><font color="Green" title="E97">A74</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1_1_1_2_1"><i><font color="Green" title="E100">A77</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:27_1_1_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E12:27"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_1"><i><font color="Green" title="E95">A72</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_1_2_1"><i><font color="Green" title="E99">A76</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2"/><i><font color="Green" title="E99">A78</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E100">A79</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_1_1_1_2_2"/><i><font color="Green" title="E101">A80</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> &amp; <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_1_2_2"><i><font color="Green" title="E99">A78</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E102">A81</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_1_1_1_2_2"/><i><font color="Green" title="E103">A82</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_1_2_2"><i><font color="Green" title="E99">A78</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_1_1_1_2_2"/><i><font color="Green" title="E104">A83</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_1_1_1_2_2"><i><font color="Green" title="E101">A80</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_1_2_2"/><i><font color="Green" title="E105">A84</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_1_1_1_2_2"><i><font color="Green" title="E100">A79</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E106">A85</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_1_2_2"><i><font color="Green" title="E99">A78</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_1_2_2"><i><font color="Green" title="E104">A83</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E9:27_1_1_1_2_2"/><i><font color="Green" title="E107">A86</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_1_2_2"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_1_2_2"><i><font color="Green" title="E103">A82</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_1_2_2"><i><font color="Green" title="E104">A83</font></i></a>, <a class="txt" href="jordan7.html#E7:27_1_1_1_2_2"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_1_1_1_2_2"/><i><font color="Green" title="E108">A87</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_1_1_1_2_2"><i><font color="Green" title="E106">A85</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_1_1_1_2_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E12:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E109">A88</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_1_1_1_2_2"/>
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E110">A89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_1_2_2"><i><font color="Green" title="E103">A82</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E111">A90</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_1_1_1_2_2"/><i><font color="Green" title="E112">A91</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E12:27_1_1_1_2_2"><i><font color="Green" title="E109">A88</font></i></a>, <a class="txt" href="jordan7.html#E14:27_1_1_1_2_2"><i><font color="Green" title="E110">A89</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E17:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E113">A92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_1_1_1_2_2"><i><font color="Green" title="E107">A86</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/><a NAME="E20:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_1"><i><font color="Green" title="E98">A75</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:27_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E114">A93</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a></i>;<br/><a NAME="E22:27_1_1_1_2_2"/><i><font color="Green" title="E115">A94</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_1_1_1_2_2"><i><font color="Green" title="E111">A90</font></i></a>, <a class="txt" href="jordan7.html#E16:27_1_1_1_2_2"><i><font color="Green" title="E112">A91</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E23:27_1_1_1_2_2"/><i><font color="Green" title="E116">A95</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_1_1_1_2_2"><i><font color="Green" title="E108">A87</font></i></a>, <a class="txt" href="jordan7.html#E16:27_1_1_1_2_2"><i><font color="Green" title="E112">A91</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_2_2_1"/>
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E21:27_1_1_1_2_2"><i><font color="Green" title="E114">A93</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_1_2_2"><i><font color="Green" title="E102">A81</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_1_2_2"><i><font color="Green" title="E103">A82</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_1_2_2"><i><font color="Green" title="E104">A83</font></i></a>, <a class="txt" href="jordan7.html#E7:27_1_1_1_2_2"><i><font color="Green" title="E105">A84</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E3:27_1_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E17:27_1_1_1_2_2"><i><font color="Green" title="E113">A92</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E22:27_1_1_1_2_2"><i><font color="Green" title="E115">A94</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E5:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E7:27_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_2_2_1_2"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E9:27_1_1_1_2_2"><i><font color="Green" title="E107">A86</font></i></a>, <a class="txt" href="jordan7.html#E17:27_1_1_1_2_2"><i><font color="Green" title="E113">A92</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_1_1_1_2_2"><i><font color="Green" title="E116">A95</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/><i><font color="Green" title="E95">A96</font></i>: 
<font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b><i><font color="Green" title="E96">A97</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_1_1_2"/><i><font color="Green" title="E97">A98</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font> &amp; <font color="Maroon" title="c12">nx</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1"><i><font color="Green" title="E93">A70</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_2"><i><font color="Green" title="E95">A96</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_1_1_2"/><b>then </b><i><font color="Green" title="E98">A99</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_1_1_2"/><i><font color="Green" title="E99">A100</font></i>: 
<font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_2"><i><font color="Green" title="E95">A96</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_1_1_2"/><i><font color="Green" title="E100">A101</font></i>: 
<font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_1_1_2"><i><font color="Green" title="E97">A98</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_2"/><i><font color="Green" title="E101">A102</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_1_1_2"><i><font color="Green" title="E96">A97</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_1_1_2"/><b>then </b><i><font color="Green" title="E102">A103</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_2"><i><font color="Green" title="E95">A96</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_2"><i><font color="Green" title="E100">A101</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E9:27_1_1_2"/><i><font color="Green" title="E103">A104</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_2"><i><font color="Green" title="E98">A99</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_2"><i><font color="Green" title="E99">A100</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_2"><i><font color="Green" title="E100">A101</font></i></a>, <a class="txt" href="jordan7.html#E7:27_1_1_2"><i><font color="Green" title="E101">A102</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_1_1_2"/><i><font color="Green" title="E104">A105</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_1_1_2"><i><font color="Green" title="E102">A103</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E12:27_1_1_2"/><b>then </b><i><font color="Green" title="E105">A106</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_1_1_2"/>
<font color="Maroon" title="c12">nx</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1"><i><font color="Green" title="E93">A70</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_1_1_2"/><b>then </b><i><font color="Green" title="E106">A107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_2"><i><font color="Green" title="E99">A100</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:27_1_1_2"/><b>then </b><i><font color="Green" title="E107">A108</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_1_1_2"/><i><font color="Green" title="E108">A109</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E12:27_1_1_2"><i><font color="Green" title="E105">A106</font></i></a>, <a class="txt" href="jordan7.html#E14:27_1_1_2"><i><font color="Green" title="E106">A107</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E17:27_1_1_2"/><b>then </b><i><font color="Green" title="E109">A110</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_1_1_2"><i><font color="Green" title="E103">A104</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_1_1_2"/><b>then </b><i><font color="Green" title="E110">A111</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">nx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_2_1"/>
( <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1"/>
<font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E111">A112</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E112">A113</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> = 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font>
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_2_1_1"><i><font color="Green" title="E111">A112</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">ny</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E2:27_1_1_2_1_1"><i><font color="Green" title="E111">A112</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:27_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1_1_2_1_1"><i><font color="Green" title="E112">A113</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E19:27_1_1_2"><i><font color="Green" title="E110">A111</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:27_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E113">A114</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a></i>;<br/><a NAME="E8:27_1_1_2_1_1"/><i><font color="Green" title="E114">A115</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_1_1_2"><i><font color="Green" title="E107">A108</font></i></a>, <a class="txt" href="jordan7.html#E16:27_1_1_2"><i><font color="Green" title="E108">A109</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E9:27_1_1_2_1_1"/><i><font color="Green" title="E115">A116</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_1_1_2"><i><font color="Green" title="E104">A105</font></i></a>, <a class="txt" href="jordan7.html#E16:27_1_1_2"><i><font color="Green" title="E108">A109</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_2_1_1_2"/>
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E7:27_1_1_2_1_1"><i><font color="Green" title="E113">A114</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_2_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E9:27_1_1_2"><i><font color="Green" title="E103">A104</font></i></a>, <a class="txt" href="jordan7.html#E17:27_1_1_2"><i><font color="Green" title="E109">A110</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E8:27_1_1_2_1_1"><i><font color="Green" title="E114">A115</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E6:27_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_2"><i><font color="Green" title="E98">A99</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_2"><i><font color="Green" title="E99">A100</font></i></a>, <a class="txt" href="jordan7.html#E6:27_1_1_2"><i><font color="Green" title="E100">A101</font></i></a>, <a class="txt" href="jordan7.html#E7:27_1_1_2"><i><font color="Green" title="E101">A102</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E3:27_1_1_2_1_1_2_2"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E17:27_1_1_2"><i><font color="Green" title="E109">A110</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:27_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_1_1_2_1_1"><i><font color="Green" title="E115">A116</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_2"/><i><font color="Green" title="E111">A117</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font></span><br/><div class="add"><a NAME="E2:27_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E112">A118</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E113">A119</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_1_1_2_1_2"/><i><font color="Green" title="E114">A120</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1_1_2_1_2"><i><font color="Green" title="E111">A117</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:27_1_1_2_1_2"/><i><font color="Green" title="E115">A121</font></i>: 
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_1_1_2_1_2"><i><font color="Green" title="E112">A118</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E116">A122</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ny</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_1_1_2_1_2"><i><font color="Green" title="E113">A119</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_2_1_2"><i><font color="Green" title="E114">A120</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_2_1_2"><i><font color="Green" title="E115">A121</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_1_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E9:27_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E117">A123</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_1_1_2_1_2"/>
<font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1"><i><font color="Green" title="E94">A71</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E11:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_2_1_2"><i><font color="Green" title="E114">A120</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:27_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E118">A124</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E9:27_1_1_2_1_2"><i><font color="Green" title="E117">A123</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E13:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E1:27_1"><i><font color="Green" title="E92">A69</font></i></a>, <a class="txt" href="jordan7.html#E9:27_1_1_2"><i><font color="Green" title="E103">A104</font></i></a>, <a class="txt" href="jordan7.html#E17:27_1_1_2"><i><font color="Green" title="E109">A110</font></i></a>, <a class="txt" href="jordan7.html#E7:27_1_1_2_1_2"><i><font color="Green" title="E116">A122</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E15:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E16:27_1_1_2"><i><font color="Green" title="E108">A109</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E16:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">nx</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_1_1_2_1_2"><i><font color="Green" title="E118">A124</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E17:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">nx</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">ny</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_1_1_2"><i><font color="Green" title="E99">A100</font></i></a>, <a class="txt" href="jordan7.html#E4:27_1_1_2_1_2"><i><font color="Green" title="E114">A120</font></i></a>, <a class="ref" href="xcmplx_1.html#T29" title="XCMPLX_1:th.29">XCMPLX_1:29</a></i>;<br/><b>hence </b><a NAME="E18:27_1_1_2_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<a NAME="E84:27"/><b>then </b><i><font color="Green" title="E92">A125</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green" title="E47">A24</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green" title="E77">A54</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E85:27"/>
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E86:27"/><b>then </b><i><font color="Green" title="E93">A126</font></i>: 
5 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E87:27"/>
8 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<a NAME="E88:27"/><i><font color="Green" title="E94">A127</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E89:27"/><i><font color="Green" title="E95">A128</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T28" title="JORDAN3:th.28">JORDAN3:28</a></i>;<br/>
<a NAME="E90:27"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E91:27"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E89:27"><i><font color="Green" title="E95">A128</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E92:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E88:27"><i><font color="Green" title="E94">A127</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E93:27"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<b>hence </b><a NAME="E94:27"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">h0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<b>thus </b><a NAME="E95:27"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E96">A129</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E2:27_2"/><b>then </b><i><font color="Green" title="E97">A130</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:27_2"/><i><font color="Green" title="E98">A131</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:27_2"/><i><font color="Green" title="E99">A132</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1"/><i><font color="Green" title="E100">A133</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_2_1_1"/><b>then </b><i><font color="Green" title="E101">A134</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_2_1_1"/><i><font color="Green" title="E102">A135</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_1"><i><font color="Green" title="E100">A133</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_2_1_1"/><i><font color="Green" title="E103">A136</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_1"><i><font color="Green" title="E100">A133</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_2_1_1"/><b>then </b><i><font color="Green" title="E104">A137</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_2_1_1"><i><font color="Green" title="E102">A135</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_2_1_1"/><i><font color="Green" title="E105">A138</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_2_1_1"><i><font color="Green" title="E103">A136</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_2_1_1"/><b>then </b><i><font color="Green" title="E106">A139</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_2_1_1"/><i><font color="Green" title="E107">A140</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_2"><i><font color="Green" title="E98">A131</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2_1_1"><i><font color="Green" title="E101">A134</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E9:27_2_1_1"/><i><font color="Green" title="E108">A141</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_2"><i><font color="Green" title="E98">A131</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2_1_1"><i><font color="Green" title="E101">A134</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:27_2_1_1"/><b>then </b><i><font color="Green" title="E109">A142</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_2_1_1"><i><font color="Green" title="E107">A140</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E11:27_2_1_1"/><i><font color="Green" title="E110">A143</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_2_1_1"><i><font color="Green" title="E108">A141</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:27_2_1_1"/><b>then </b><i><font color="Green" title="E111">A144</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E13:27_2_1_1"/><i><font color="Green" title="E112">A145</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E4:27_2"><i><font color="Green" title="E99">A132</font></i></a>, <a class="txt" href="jordan7.html#E4:27_2_1_1"><i><font color="Green" title="E103">A136</font></i></a>, <a class="txt" href="jordan7.html#E9:27_2_1_1"><i><font color="Green" title="E108">A141</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E14:27_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:27_2_1_1"/><b>then </b><i><font color="Green" title="E113">A146</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E16:27_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="txt" href="jordan7.html#E3:27_2"><i><font color="Green" title="E98">A131</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:27_2_1_1"/><b>then </b><i><font color="Green" title="E114">A147</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E18:27_2_1_1"/><i><font color="Green" title="E115">A148</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E6:27_2_1_1"><i><font color="Green" title="E105">A138</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_2_1_1"/><i><font color="Green" title="E116">A149</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E10:27_2_1_1"><i><font color="Green" title="E109">A142</font></i></a>, <a class="txt" href="jordan7.html#E11:27_2_1_1"><i><font color="Green" title="E110">A143</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_2_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_2_1_1"><i><font color="Green" title="E104">A137</font></i></a>, <a class="txt" href="jordan7.html#E7:27_2_1_1"><i><font color="Green" title="E106">A139</font></i></a>, <a class="txt" href="jordan7.html#E10:27_2_1_1"><i><font color="Green" title="E109">A142</font></i></a>, <a class="txt" href="jordan7.html#E12:27_2_1_1"><i><font color="Green" title="E111">A144</font></i></a>, <a class="txt" href="jordan7.html#E13:27_2_1_1"><i><font color="Green" title="E112">A145</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_1"><i><font color="Green" title="E113">A146</font></i></a>, <a class="txt" href="jordan7.html#E17:27_2_1_1"><i><font color="Green" title="E114">A147</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E21:27_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E5:27_2_1_1"><i><font color="Green" title="E104">A137</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_1"><i><font color="Green" title="E113">A146</font></i></a>, <a class="txt" href="jordan7.html#E17:27_2_1_1"><i><font color="Green" title="E114">A147</font></i></a>, <a class="txt" href="jordan7.html#E18:27_2_1_1"><i><font color="Green" title="E115">A148</font></i></a>, <a class="txt" href="jordan7.html#E19:27_2_1_1"><i><font color="Green" title="E116">A149</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2"/><i><font color="Green" title="E100">A150</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E1:27_2_1_2"><i><font color="Green" title="E100">A150</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_2_1_1"/><i><font color="Green" title="E101">A151</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E102">A152</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_1"><i><font color="Green" title="E101">A151</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E103">A153</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_2_1_2_1_1"/><i><font color="Green" title="E104">A154</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_1"><i><font color="Green" title="E101">A151</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_2_1_2_1_1"/><i><font color="Green" title="E105">A155</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_2_1_2_1_1"><i><font color="Green" title="E102">A152</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E106">A156</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_2_1_2_1_1"><i><font color="Green" title="E103">A153</font></i></a>, <a class="txt" href="jordan7.html#E5:27_2_1_2_1_1"><i><font color="Green" title="E104">A154</font></i></a>, <a class="txt" href="jordan7.html#E6:27_2_1_2_1_1"><i><font color="Green" title="E105">A155</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_2_1_2_1_1"/><i><font color="Green" title="E107">A157</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_2_1_2_1_1"><i><font color="Green" title="E102">A152</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E108">A158</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_2_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_1"><i><font color="Green" title="E101">A151</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E109">A159</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E13:27_2_1_2_1_1"/><i><font color="Green" title="E110">A160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_2_1_2_1_1"/><i><font color="Green" title="E111">A161</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_2_1_2_1_1"><i><font color="Green" title="E107">A157</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E112">A162</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E10:27_2_1_2_1_1"><i><font color="Green" title="E108">A158</font></i></a>, <a class="txt" href="jordan7.html#E12:27_2_1_2_1_1"><i><font color="Green" title="E109">A159</font></i></a>, <a class="txt" href="jordan7.html#E13:27_2_1_2_1_1"><i><font color="Green" title="E110">A160</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_2_1_2_1_1"/><i><font color="Green" title="E113">A163</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_2_1_2_1_1"><i><font color="Green" title="E109">A159</font></i></a>, <a class="txt" href="jordan7.html#E14:27_2_1_2_1_1"><i><font color="Green" title="E111">A161</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E114">A164</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E18:27_2_1_2_1_1"/><i><font color="Green" title="E115">A165</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_2_1_2_1_1"><i><font color="Green" title="E113">A163</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E19:27_2_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E20:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E116">A166</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_2_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E117">A167</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_2_1_2_1_1"><i><font color="Green" title="E104">A154</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E23:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E118">A168</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E20:27_2_1_2_1_1"><i><font color="Green" title="E116">A166</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><i><font color="Green" title="E119">A169</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E23:27_2_1_2_1_1"><i><font color="Green" title="E118">A168</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2

;<br/><i><font color="Green" title="E120">A170</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_2_1_2_1_1"><i><font color="Green" title="E104">A154</font></i></a>, <a class="txt" href="jordan7.html#E23:27_2_1_2_1_1"><i><font color="Green" title="E118">A168</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E12:27_2_1_2_1_1"><i><font color="Green" title="E109">A159</font></i></a>, <a class="txt" href="jordan7.html#E17:27_2_1_2_1_1"><i><font color="Green" title="E114">A164</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E26:27_2_1_2_1_1"/><i><font color="Green" title="E121">A171</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E23:27_2_1_2_1_1"><i><font color="Green" title="E118">A168</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E122">A172</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_2_1_2_1_1"><i><font color="Green" title="E106">A156</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E28:27_2_1_2_1_1"/><i><font color="Green" title="E123">A173</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27_2_1_2_1_1"><i><font color="Green" title="E121">A171</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E124">A174</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E30:27_2_1_2_1_1"/><i><font color="Green" title="E125">A175</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E31:27_2_1_2_1_1"/><i><font color="Green" title="E126">A176</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_2_1_2_1_1"><i><font color="Green" title="E118">A168</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E32:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E127">A177</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27_2_1_2_1_1"><i><font color="Green" title="E117">A167</font></i></a>, <a class="txt" href="jordan7.html#E24:27_2_1_2_1_1"><i><font color="Green" title="E119">A169</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E33:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E128">A178</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_2_1_2_1_1"><i><font color="Green" title="E112">A162</font></i></a>, <a class="txt" href="jordan7.html#E25:27_2_1_2_1_1"><i><font color="Green" title="E120">A170</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E34:27_2_1_2_1_1"/><i><font color="Green" title="E129">A179</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27_2_1_2_1_1"><i><font color="Green" title="E127">A177</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E35:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E130">A180</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E36:27_2_1_2_1_1"/><i><font color="Green" title="E131">A181</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E26:27_2_1_2_1_1"><i><font color="Green" title="E121">A171</font></i></a>, <a class="txt" href="jordan7.html#E30:27_2_1_2_1_1"><i><font color="Green" title="E125">A175</font></i></a>, <a class="txt" href="jordan7.html#E32:27_2_1_2_1_1"><i><font color="Green" title="E127">A177</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E37:27_2_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E38:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E132">A182</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E39:27_2_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="txt" href="jordan7.html#E3:27_2"><i><font color="Green" title="E98">A131</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E40:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E133">A183</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E41:27_2_1_2_1_1"/><i><font color="Green" title="E134">A184</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E27:27_2_1_2_1_1"><i><font color="Green" title="E122">A172</font></i></a>, <a class="txt" href="jordan7.html#E28:27_2_1_2_1_1"><i><font color="Green" title="E123">A173</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E42:27_2_1_2_1_1"/><i><font color="Green" title="E135">A185</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E33:27_2_1_2_1_1"><i><font color="Green" title="E128">A178</font></i></a>, <a class="txt" href="jordan7.html#E34:27_2_1_2_1_1"><i><font color="Green" title="E129">A179</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E43:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E44:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E136">A186</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_2_1_2_1_1"><i><font color="Green" title="E109">A159</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E45:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E137">A187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E46:27_2_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E47:27_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E48:27_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27_2_1_2_1_1"><i><font color="Green" title="E115">A165</font></i></a>, <a class="txt" href="jordan7.html#E45:27_2_1_2_1_1"><i><font color="Green" title="E137">A187</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E49:27_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E138">A188</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27_2_1_2_1_1"><i><font color="Green" title="E120">A170</font></i></a>, <a class="txt" href="jordan7.html#E31:27_2_1_2_1_1"><i><font color="Green" title="E126">A176</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><i><font color="Green" title="E139">A189</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1_2_1_1_3"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_2_1_2_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_2_1_1"><i><font color="Green" title="E112">A162</font></i></a>, <a class="txt" href="jordan7.html#E40:27_2_1_2_1_1"><i><font color="Green" title="E133">A183</font></i></a>, <a class="txt" href="jordan7.html#E49:27_2_1_2_1_1"><i><font color="Green" title="E138">A188</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_2_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E18:27_2_1_2_1_1"><i><font color="Green" title="E115">A165</font></i></a>, <a class="txt" href="jordan7.html#E44:27_2_1_2_1_1"><i><font color="Green" title="E136">A186</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:27_2_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E5:27_2_1_2_1_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E51:27_2_1_2_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E27:27_2_1_2_1_1"><i><font color="Green" title="E122">A172</font></i></a>, <a class="txt" href="jordan7.html#E29:27_2_1_2_1_1"><i><font color="Green" title="E124">A174</font></i></a>, <a class="txt" href="jordan7.html#E33:27_2_1_2_1_1"><i><font color="Green" title="E128">A178</font></i></a>, <a class="txt" href="jordan7.html#E35:27_2_1_2_1_1"><i><font color="Green" title="E130">A180</font></i></a>, <a class="txt" href="jordan7.html#E36:27_2_1_2_1_1"><i><font color="Green" title="E131">A181</font></i></a>, <a class="txt" href="jordan7.html#E38:27_2_1_2_1_1"><i><font color="Green" title="E132">A182</font></i></a>, <a class="txt" href="jordan7.html#E40:27_2_1_2_1_1"><i><font color="Green" title="E133">A183</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E52:27_2_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27_2_1_2_1_1"><i><font color="Green" title="E132">A182</font></i></a>, <a class="txt" href="jordan7.html#E40:27_2_1_2_1_1"><i><font color="Green" title="E133">A183</font></i></a>, <a class="txt" href="jordan7.html#E41:27_2_1_2_1_1"><i><font color="Green" title="E134">A184</font></i></a>, <a class="txt" href="jordan7.html#E42:27_2_1_2_1_1"><i><font color="Green" title="E135">A185</font></i></a>, <a class="txt" href="jordan7.html#E50:27_2_1_2_1_1"><i><font color="Green" title="E139">A189</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2_1_2"/><i><font color="Green" title="E101">A190</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E102">A191</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E3:27_2_1_2_1_2"/><i><font color="Green" title="E103">A192</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_2"><i><font color="Green" title="E99">A132</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:27_2_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E104">A193</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="txt" href="jordan7.html#E2:27_2_1_2_1_2"><i><font color="Green" title="E102">A191</font></i></a>, <a class="txt" href="jordan7.html#E3:27_2_1_2_1_2"><i><font color="Green" title="E103">A192</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E6:27_2_1_2_1_2"/><i><font color="Green" title="E105">A194</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_2_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E106">A195</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="txt" href="jordan7.html#E6:27_2_1_2_1_2"><i><font color="Green" title="E105">A194</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E9:27_2_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_2"><i><font color="Green" title="E96">A129</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:27_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_2_1_2_1_2"><i><font color="Green" title="E106">A195</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E11:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E107">A196</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/><a NAME="E13:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E108">A197</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E14:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E109">A198</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><i><font color="Green" title="E110">A199</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_2_1_2_1_2"><i><font color="Green" title="E101">A190</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a></i>
;<br/><a NAME="E16:27_2_1_2_1_2"/><i><font color="Green" title="E111">A200</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_2_1_2_1_2"><i><font color="Green" title="E108">A197</font></i></a>, <a class="txt" href="jordan7.html#E14:27_2_1_2_1_2"><i><font color="Green" title="E109">A198</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E112">A201</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_2_1_2_1_2"><i><font color="Green" title="E104">A193</font></i></a>, <a class="txt" href="jordan7.html#E13:27_2_1_2_1_2"><i><font color="Green" title="E108">A197</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_2_1_2"><i><font color="Green" title="E110">A199</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E18:27_2_1_2_1_2"/><i><font color="Green" title="E113">A202</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_2_1_2_1_2"><i><font color="Green" title="E111">A200</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_2_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_2"><i><font color="Green" title="E97">A130</font></i></a>, <a class="txt" href="jordan7.html#E3:27_2"><i><font color="Green" title="E98">A131</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E20:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E114">A203</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E21:27_2_1_2_1_2"/><i><font color="Green" title="E115">A204</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E17:27_2_1_2_1_2"><i><font color="Green" title="E112">A201</font></i></a>, <a class="txt" href="jordan7.html#E18:27_2_1_2_1_2"><i><font color="Green" title="E113">A202</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:27_2_1_2_1_2"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E23:27_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E116">A205</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_2_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_2_1_2"><i><font color="Green" title="E110">A199</font></i></a></i>;<br/><a NAME="E25:27_2_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E5:27_2_1_2_1_2"><i><font color="Green" title="E104">A193</font></i></a>, <a class="txt" href="jordan7.html#E15:27_2_1_2_1_2"><i><font color="Green" title="E110">A199</font></i></a>, <a class="txt" href="jordan7.html#E20:27_2_1_2_1_2"><i><font color="Green" title="E114">A203</font></i></a>, <a class="txt" href="jordan7.html#E23:27_2_1_2_1_2"><i><font color="Green" title="E116">A205</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E26:27_2_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_2_1_2_1_2"><i><font color="Green" title="E107">A196</font></i></a>, <a class="txt" href="jordan7.html#E20:27_2_1_2_1_2"><i><font color="Green" title="E114">A203</font></i></a>, <a class="txt" href="jordan7.html#E21:27_2_1_2_1_2"><i><font color="Green" title="E115">A204</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E96:27"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b2">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/><b>let </b><font color="Maroon" title="c11">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> implies  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> )</span><br/>



<b>assume </b><a NAME="E1:27_3"/><i><font color="Green" title="E96">A206</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/>

<a NAME="E2:27_3"/><b>then </b><i><font color="Green" title="E97">A207</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:27_3"/><i><font color="Green" title="E98">A208</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:27_3"/><i><font color="Green" title="E99">A209</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1"/><i><font color="Green" title="E100">A210</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/><div class="add"><a NAME="E2:27_3_1_1"/><b>then </b><i><font color="Green" title="E101">A211</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_3_1_1"/><i><font color="Green" title="E102">A212</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_3_1_1"/><i><font color="Green" title="E103">A213</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_3_1_1"/><b>then </b><i><font color="Green" title="E104">A214</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_3_1_1"/><b>then </b><i><font color="Green" title="E105">A215</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_3_1_1"/><i><font color="Green" title="E106">A216</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1"><i><font color="Green" title="E103">A213</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1"/><b>then </b><i><font color="Green" title="E107">A217</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1"/><i><font color="Green" title="E108">A218</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E10:27_3_1_1"/><i><font color="Green" title="E109">A219</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:27_3_1_1"/><b>then </b><i><font color="Green" title="E110">A220</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_3_1_1"><i><font color="Green" title="E108">A218</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E12:27_3_1_1"/><i><font color="Green" title="E111">A221</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:27_3_1_1"/><b>then </b><i><font color="Green" title="E112">A222</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_3_1_1"/><i><font color="Green" title="E113">A223</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3"><i><font color="Green" title="E99">A209</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1"><i><font color="Green" title="E103">A213</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E15:27_3_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_3_1_1"/><b>then </b><i><font color="Green" title="E114">A224</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E17:27_3_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_3_1_1"/><b>then </b><i><font color="Green" title="E115">A225</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E19:27_3_1_1"/><i><font color="Green" title="E116">A226</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1"><i><font color="Green" title="E105">A215</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_1"><i><font color="Green" title="E106">A216</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_3_1_1"/><i><font color="Green" title="E117">A227</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1"><i><font color="Green" title="E110">A220</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_1"><i><font color="Green" title="E111">A221</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:27_3_1_1"/><i><font color="Green" title="E118">A228</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E22:27_3_1_1"/><i><font color="Green" title="E119">A229</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_1"><i><font color="Green" title="E106">A216</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_1"><i><font color="Green" title="E111">A221</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><a NAME="E24:27_3_1_1"/><i><font color="Green" title="E120">A230</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_1_1"/><i><font color="Green" title="E121">A231</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:27_3_1_1_1"/><i><font color="Green" title="E122">A232</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1"><i><font color="Green" title="E108">A218</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3_1_1_1"><i><font color="Green" title="E121">A231</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_1_1"/><i><font color="Green" title="E123">A233</font></i>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_1_1"/><i><font color="Green" title="E124">A234</font></i>: 
( ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c14">p</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c14">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_1_1"/><i><font color="Green" title="E125">A235</font></i>: 
( ( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1"/>
( <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E126">A236</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green" title="E49">A26</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1"><i><font color="Green" title="E108">A218</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><div><i><font color="Green" title="E127">A237</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_1"><i><font color="Green" title="E117">A227</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3_1_1_1"><i><font color="Green" title="E122">A232</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_1"><i><font color="Green" title="E126">A236</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E4:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E128">A238</font></i>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><a NAME="E5:27_3_1_1_1_1_1_1"/><i><font color="Green" title="E129">A239</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_1_1_1_1"><i><font color="Green" title="E127">A237</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><a NAME="E6:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E130">A240</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_1"><i><font color="Green" title="E126">A236</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E131">A241</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1"><i><font color="Green" title="E103">A213</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1"><i><font color="Green" title="E104">A214</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1"><i><font color="Green" title="E105">A215</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E132">A242</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green" title="E49">A26</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1"><i><font color="Green" title="E103">A213</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="txt" href="jordan7.html#E19:27_3_1_1"><i><font color="Green" title="E116">A226</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_1"><i><font color="Green" title="E102">A212</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E131">A241</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E132">A242</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E133">A243</font></i>: 
( ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c14">p</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><a NAME="E6:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E134">A244</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E131">A241</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/><div><i><font color="Green" title="E135">A245</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1"><i><font color="Green" title="E128">A238</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_3_1_1_1_1_1_1"><i><font color="Green" title="E130">A240</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E134">A244</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E136">A246</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E133">A243</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">rz</font> = <font color="Maroon" title="c15">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E137">A247</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1"><i><font color="Green" title="E102">A212</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1"><i><font color="Green" title="E104">A214</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E12:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E138">A248</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1"><i><font color="Green" title="E107">A217</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1"><i><font color="Green" title="E110">A220</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E13:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E139">A249</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>take </b><font color="Maroon" title="c17">z</font> = <font color="Maroon" title="c15">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/><b>thus </b><a NAME="E15:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/><a NAME="E16:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green" title="E140">A250</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">rz</font> &amp; <font color="Maroon" title="c16">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E17:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E141">A251</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">rz</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E133">A243</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E135">A245</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E137">A247</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E138">A248</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><a NAME="E18:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c16">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1_1_1_1"><i><font color="Green" title="E129">A239</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E138">A248</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E139">A249</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E140">A250</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E19:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E141">A251</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/><b>thus </b><a NAME="E20:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_1"><i><font color="Green" title="E136">A246</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2"/><i><font color="Green" title="E131">A252</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E132">A253</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">rz</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">rz</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1"><i><font color="Green" title="E128">A238</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_3_1_1_1_1_1_1"><i><font color="Green" title="E130">A240</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E132">A253</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E133">A254</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1"><i><font color="Green" title="E124">A234</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">rz</font> = <font color="Maroon" title="c15">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E133">A254</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c17">rz</font> = <font color="Maroon" title="c16">rz</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">rz</font> )</span><br/><a NAME="E6:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E134">A255</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1"><i><font color="Green" title="E110">A220</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1_1_1_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E135">A256</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E136">A257</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E133">A254</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E10:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E137">A258</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E133">A254</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1_1_1_1"><i><font color="Green" title="E129">A239</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E133">A254</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E134">A255</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E135">A256</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E136">A257</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E12:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">rz</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E133">A254</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1_1_1_1_1_2_2_1_1_1"><i><font color="Green" title="E137">A258</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_2"/><i><font color="Green" title="E132">A259</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><div class="add"><b>thus </b><a NAME="E2:27_3_1_1_1_1_1_1_2_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_1"><i><font color="Green" title="E113">A223</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><b>thus </b><a NAME="E3:27_3_1_1_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2_1_1_2"><i><font color="Green" title="E132">A259</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_1_2_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_1_2_2"><i><font color="Green" title="E131">A252</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2"/><i><font color="Green" title="E126">A260</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E127">A261</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green" title="E49">A26</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1"><i><font color="Green" title="E108">A218</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2"/><i><font color="Green" title="E128">A262</font></i>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><div><i><font color="Green" title="E129">A263</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_2_1"/><i><font color="Green" title="E130">A264</font></i>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_2_1"/><b>then </b>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_2"><i><font color="Green" title="E127">A261</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_1_1"><i><font color="Green" title="E125">A235</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_2"><i><font color="Green" title="E127">A261</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2_1"><i><font color="Green" title="E130">A264</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:27_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E130">A265</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c14">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_2"><i><font color="Green" title="E127">A261</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_1_1_1_2"><i><font color="Green" title="E128">A262</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_2_2"/>
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1"/><i><font color="Green" title="E131">A266</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1"/>
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_1"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green" title="E132">A267</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">rz</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">rz</font> )</span><br/><div class="add"><div><i><font color="Green" title="E133">A268</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2"><i><font color="Green" title="E129">A263</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2_2_1"><i><font color="Green" title="E131">A266</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E132">A267</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green" title="E134">A269</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1"><i><font color="Green" title="E124">A234</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">rz</font> = <font color="Maroon" title="c15">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E134">A269</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c17">rz</font> = <font color="Maroon" title="c16">rz</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">rz</font> )</span><br/><a NAME="E6:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green" title="E135">A270</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1"><i><font color="Green" title="E101">A211</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1"><i><font color="Green" title="E110">A220</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E136">A271</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green" title="E137">A272</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E134">A269</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E10:27_3_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E138">A273</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1"><i><font color="Green" title="E102">A212</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1"><i><font color="Green" title="E104">A214</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1"><i><font color="Green" title="E107">A217</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1"><i><font color="Green" title="E124">A234</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E133">A268</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E134">A269</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1_1_1_2"><i><font color="Green" title="E130">A265</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E134">A269</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E135">A270</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E136">A271</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E137">A272</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E12:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">rz</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E134">A269</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1_1_1_1_2_2_1_1_1_1"><i><font color="Green" title="E138">A273</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_2"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_2"/><i><font color="Green" title="E132">A274</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1"><i><font color="Green" title="E102">A212</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1_1"><i><font color="Green" title="E124">A234</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1"><i><font color="Green" title="E103">A213</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_1"><i><font color="Green" title="E113">A223</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_1_1_1_1_2_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2_2_1_1_1_2"><i><font color="Green" title="E132">A274</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_2_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_2"/><i><font color="Green" title="E131">A275</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2_2_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_1"><i><font color="Green" title="E113">A223</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_1"><i><font color="Green" title="E118">A228</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2"><i><font color="Green" title="E126">A260</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_2_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green" title="E56">A33</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_1_1"><i><font color="Green" title="E123">A233</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1_1_1_1_2_2_2"><i><font color="Green" title="E131">A275</font></i></a>, <a class="txt" href="jordan7.html#E23"><i><font color="Green" title="E20">Lm5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:27_3_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E25:27_3_1_1"/>
<font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_1_2"/><i><font color="Green" title="E121">A276</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g1</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_1_2"/><i><font color="Green" title="E122">A277</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">p1</font> = <font color="Maroon" title="c13">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">sy</font> = <font color="Maroon" title="c14">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_1_2"/><i><font color="Green" title="E123">A278</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_1"><i><font color="Green" title="E105">A215</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:27_3_1_1_2"/><i><font color="Green" title="E124">A279</font></i>: 
( <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">sy</font> &amp; <font color="Maroon" title="c16">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_1_2"/><i><font color="Green" title="E125">A280</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1"><i><font color="Green" title="E107">A217</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:27_3_1_1_2"/><i><font color="Green" title="E126">A281</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">sy</font> &amp; <font color="Maroon" title="c16">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:27_3_1_1_2"/><b>then </b><i><font color="Green" title="E127">A282</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p1</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_1_2"><i><font color="Green" title="E123">A278</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1_2"><i><font color="Green" title="E124">A279</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1_2"><i><font color="Green" title="E125">A280</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E12:27_3_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1"><i><font color="Green" title="E110">A220</font></i></a>, <a class="txt" href="jordan7.html#E13:27_3_1_1"><i><font color="Green" title="E112">A222</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_1"><i><font color="Green" title="E119">A229</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_1_2"><i><font color="Green" title="E121">A276</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_1_2"><i><font color="Green" title="E124">A279</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1_2"><i><font color="Green" title="E126">A281</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E13:27_3_1_1_2"/><b>then </b><i><font color="Green" title="E128">A283</font></i>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c15">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27_3_1_1"><i><font color="Green" title="E114">A224</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="txt" href="jordan7.html#E19:27_3_1_1"><i><font color="Green" title="E116">A226</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_1"><i><font color="Green" title="E117">A227</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_1_2"><i><font color="Green" title="E122">A277</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_1_2"><i><font color="Green" title="E127">A282</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E14:27_3_1_1_2"/><i><font color="Green" title="E129">A284</font></i>: 
not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_1"><i><font color="Green" title="E108">A218</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_1"><i><font color="Green" title="E109">A219</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_1"><i><font color="Green" title="E115">A225</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:27_3_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27_3_1_1_2"><i><font color="Green" title="E128">A283</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:27_3_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_3_1_1_2"><i><font color="Green" title="E129">A284</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E26:27_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E24:27_3_1_1"><i><font color="Green" title="E120">A230</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E27:27_3_1_1"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_1"><i><font color="Green" title="E100">A210</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_2"/><i><font color="Green" title="E100">A285</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/><div class="add"><a NAME="E2:27_3_1_2"/><b>then </b><i><font color="Green" title="E101">A286</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_3_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_2"><i><font color="Green" title="E100">A285</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_3_1_2"/><b>then </b><i><font color="Green" title="E102">A287</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_3_1_2"/><i><font color="Green" title="E103">A288</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_2"><i><font color="Green" title="E100">A285</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_3_1_2"/><i><font color="Green" title="E104">A289</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3_1_2"><i><font color="Green" title="E101">A286</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_3_1_2"/><b>then </b><i><font color="Green" title="E105">A290</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_2"><i><font color="Green" title="E102">A287</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_2"><i><font color="Green" title="E104">A289</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_3_1_2"/><i><font color="Green" title="E106">A291</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3_1_2"><i><font color="Green" title="E101">A286</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:27_3_1_2"/><b>then </b><i><font color="Green" title="E107">A292</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_3_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_2"><i><font color="Green" title="E100">A285</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:27_3_1_2"/><b>then </b><i><font color="Green" title="E108">A293</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E13:27_3_1_2"/><i><font color="Green" title="E109">A294</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_3_1_2"/><i><font color="Green" title="E110">A295</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_3_1_2"><i><font color="Green" title="E106">A291</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_3_1_2"/><b>then </b><i><font color="Green" title="E111">A296</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_2"><i><font color="Green" title="E107">A292</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_2"><i><font color="Green" title="E108">A293</font></i></a>, <a class="txt" href="jordan7.html#E13:27_3_1_2"><i><font color="Green" title="E109">A294</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_3_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_2"><i><font color="Green" title="E108">A293</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_2"><i><font color="Green" title="E110">A295</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_3_1_2"/><b>then </b><i><font color="Green" title="E112">A297</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E18:27_3_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E19:27_3_1_2"/><b>then </b><i><font color="Green" title="E113">A298</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:27_3_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E21:27_3_1_2"/><b>then </b><i><font color="Green" title="E114">A299</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E22:27_3_1_2"/><b>then </b><i><font color="Green" title="E115">A300</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_3_1_2"/><i><font color="Green" title="E116">A301</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E19:27_3_1_2"><i><font color="Green" title="E113">A298</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_2"><i><font color="Green" title="E114">A299</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E24:27_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E25:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E26:27_3_1_2"/><b>then </b><i><font color="Green" title="E117">A302</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27_3_1_2"><i><font color="Green" title="E115">A300</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E27:27_3_1_2"/><i><font color="Green" title="E118">A303</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c12">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E28:27_3_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3_1_2"><i><font color="Green" title="E100">A285</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E29:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E30:27_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E31:27_3_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E32:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_3_1_2"/><b>then </b><i><font color="Green" title="E119">A304</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27_3_1_2"><i><font color="Green" title="E118">A303</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E34:27_3_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E35:27_3_1_2"/><b>then </b><i><font color="Green" title="E120">A305</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E36:27_3_1_2"/><i><font color="Green" title="E121">A306</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="txt" href="jordan7.html#E26:27_3_1_2"><i><font color="Green" title="E117">A302</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><i><font color="Green" title="E122">A307</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E38:27_3_1_2"/><i><font color="Green" title="E123">A308</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_2"><i><font color="Green" title="E108">A293</font></i></a>, <a class="txt" href="jordan7.html#E17:27_3_1_2"><i><font color="Green" title="E112">A297</font></i></a>, <a class="txt" href="jordan7.html#E27:27_3_1_2"><i><font color="Green" title="E118">A303</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E39:27_3_1_2"/><i><font color="Green" title="E124">A309</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E40:27_3_1_2"/><b>then </b><i><font color="Green" title="E125">A310</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_3_1_2"><i><font color="Green" title="E105">A290</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E41:27_3_1_2"/><i><font color="Green" title="E126">A311</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27_3_1_2"><i><font color="Green" title="E124">A309</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E42:27_3_1_2"/><i><font color="Green" title="E127">A312</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E43:27_3_1_2"/><b>then </b><i><font color="Green" title="E128">A313</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27_3_1_2"><i><font color="Green" title="E114">A299</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E44:27_3_1_2"/><b>then </b><i><font color="Green" title="E129">A314</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E45:27_3_1_2"/><i><font color="Green" title="E130">A315</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E46:27_3_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E47:27_3_1_2"/><b>then </b><i><font color="Green" title="E131">A316</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E48:27_3_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E49:27_3_1_2"/><b>then </b><i><font color="Green" title="E132">A317</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E50:27_3_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_2"><i><font color="Green" title="E126">A311</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E51:27_3_1_2"/><b>then </b><i><font color="Green" title="E133">A318</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2"><i><font color="Green" title="E105">A290</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_2"><i><font color="Green" title="E124">A309</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E52:27_3_1_2"/><i><font color="Green" title="E134">A319</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E44:27_3_1_2"><i><font color="Green" title="E129">A314</font></i></a>, <a class="txt" href="jordan7.html#E45:27_3_1_2"><i><font color="Green" title="E130">A315</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E53:27_3_1_2"/><i><font color="Green" title="E135">A320</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_2"><i><font color="Green" title="E126">A311</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E54:27_3_1_2"/><i><font color="Green" title="E136">A321</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_2"><i><font color="Green" title="E126">A311</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E55:27_3_1_2"/><i><font color="Green" title="E137">A322</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E56:27_3_1_2"/><i><font color="Green" title="E138">A323</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E45:27_3_1_2"><i><font color="Green" title="E130">A315</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E57:27_3_1_2"/><i><font color="Green" title="E139">A324</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="txt" href="jordan7.html#E36:27_3_1_2"><i><font color="Green" title="E121">A306</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E58:27_3_1_2"/><i><font color="Green" title="E140">A325</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27_3_1_2"><i><font color="Green" title="E114">A299</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="txt" href="jordan7.html#E42:27_3_1_2"><i><font color="Green" title="E127">A312</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_2"><i><font color="Green" title="E126">A311</font></i></a>, <a class="txt" href="jordan7.html#E45:27_3_1_2"><i><font color="Green" title="E130">A315</font></i></a>, <a class="txt" href="jordan7.html#E57:27_3_1_2"><i><font color="Green" title="E139">A324</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><a NAME="E60:27_3_1_2"/><i><font color="Green" title="E141">A326</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_2_2"/><i><font color="Green" title="E142">A327</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:27_3_1_2_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_2"><i><font color="Green" title="E120">A305</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_2_2"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3_1_2_2"><i><font color="Green" title="E142">A327</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_2_2"/><i><font color="Green" title="E143">A328</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_2_2"/><i><font color="Green" title="E144">A329</font></i>: 
( ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2_2"><i><font color="Green" title="E143">A328</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_2_2"/><i><font color="Green" title="E145">A330</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E21:27_3_1_2"><i><font color="Green" title="E114">A299</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:27_3_1_2_2"/><i><font color="Green" title="E146">A331</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E42:27_3_1_2"><i><font color="Green" title="E127">A312</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E9:27_3_1_2_2"/><i><font color="Green" title="E147">A332</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_2"><i><font color="Green" title="E120">A305</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_1"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="txt" href="jordan7.html#E52:27_3_1_2"><i><font color="Green" title="E134">A319</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2_2"><i><font color="Green" title="E146">A331</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_2_2"><i><font color="Green" title="E147">A332</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_2_2"/><b>then </b><i><font color="Green" title="E148">A333</font></i>: 
( ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2_2"><i><font color="Green" title="E143">A328</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E12:27_3_1_2_2"/><i><font color="Green" title="E149">A334</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2"><i><font color="Green" title="E105">A290</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_2"><i><font color="Green" title="E124">A309</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_2_2"><i><font color="Green" title="E145">A330</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E13:27_3_1_2_2"/><i><font color="Green" title="E150">A335</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_2"><i><font color="Green" title="E119">A304</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_2"><i><font color="Green" title="E124">A309</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E151">A336</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E51:27_3_1_2"><i><font color="Green" title="E133">A318</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_3_1_2"><i><font color="Green" title="E105">A290</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_2_2"><i><font color="Green" title="E149">A334</font></i></a>, <a class="txt" href="jordan7.html#E13:27_3_1_2_2"><i><font color="Green" title="E150">A335</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E15:27_3_1_2_2"/><b>then </b><i><font color="Green" title="E152">A337</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2_2"><i><font color="Green" title="E143">A328</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:27_3_1_2_2"/><i><font color="Green" title="E153">A338</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2"><i><font color="Green" title="E105">A290</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_2"><i><font color="Green" title="E119">A304</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_2"><i><font color="Green" title="E124">A309</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E154">A339</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_3"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_3_1_2_2"><i><font color="Green" title="E144">A329</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_2_2"><i><font color="Green" title="E151">A336</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_2_2_3"/><b>then </b>
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_2_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_2_2"><i><font color="Green" title="E144">A329</font></i></a>, <a class="txt" href="jordan7.html#E14:27_3_1_2_2"><i><font color="Green" title="E151">A336</font></i></a>, <a class="txt" href="jordan7.html#E16:27_3_1_2_2"><i><font color="Green" title="E153">A338</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E18:27_3_1_2_2"/><b>then </b><i><font color="Green" title="E155">A340</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2_2"><i><font color="Green" title="E143">A328</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:27_3_1_2_2"/><i><font color="Green" title="E156">A341</font></i>: 
( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_2_2"><i><font color="Green" title="E148">A333</font></i></a>, <a class="txt" href="jordan7.html#E17:27_3_1_2_2"><i><font color="Green" title="E154">A339</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">rz</font> = <font color="Maroon" title="c16">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_2_2"><i><font color="Green" title="E156">A341</font></i></a></i>;<br/>
<a NAME="E22:27_3_1_2_2"/><i><font color="Green" title="E157">A342</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="txt" href="jordan7.html#E58:27_3_1_2"><i><font color="Green" title="E140">A325</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E23:27_3_1_2_2"/><i><font color="Green" title="E158">A343</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_2"><i><font color="Green" title="E126">A311</font></i></a>, <a class="txt" href="jordan7.html#E57:27_3_1_2"><i><font color="Green" title="E139">A324</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E24:27_3_1_2_2"/>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:27_3_1_2_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E58:27_3_1_2"><i><font color="Green" title="E140">A325</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_2_2"><i><font color="Green" title="E156">A341</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:27_3_1_2_2"/><b>then </b>
( <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E40:27_3_1_2"><i><font color="Green" title="E125">A310</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E57:27_3_1_2"><i><font color="Green" title="E139">A324</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_2_2"><i><font color="Green" title="E152">A337</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_2_2"><i><font color="Green" title="E155">A340</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_2_2"><i><font color="Green" title="E156">A341</font></i></a>, <a class="txt" href="jordan7.html#E22:27_3_1_2_2"><i><font color="Green" title="E157">A342</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_2_2"><i><font color="Green" title="E158">A343</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/>
<a NAME="E27:27_3_1_2_2"/><b>then </b>
<font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E28:27_3_1_2_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_2_2"><i><font color="Green" title="E143">A328</font></i></a>, <a class="txt" href="jordan7.html#E20:27_3_1_2_2"><i><font color="Green" title="E156">A341</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E61:27_3_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_2_3"/><i><font color="Green" title="E142">A344</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_2_3"/><i><font color="Green" title="E143">A345</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">p1</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">sy</font> = <font color="Maroon" title="c15">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_2_3"/><i><font color="Green" title="E144">A346</font></i>: 
( <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_3_1_2_3"/><i><font color="Green" title="E145">A347</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_2_3"/><i><font color="Green" title="E146">A348</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_2"><i><font color="Green" title="E120">A305</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E147">A349</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_3_1"/>
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_2_3_1"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">sy</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green" title="E55">A32</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E56:27_3_1_2"><i><font color="Green" title="E138">A323</font></i></a>, <a class="txt" href="jordan7.html#E58:27_3_1_2"><i><font color="Green" title="E140">A325</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_2_3"><i><font color="Green" title="E144">A346</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_2_3"><i><font color="Green" title="E146">A348</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_2_3"/><i><font color="Green" title="E148">A350</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_2"><i><font color="Green" title="E103">A288</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_2"><i><font color="Green" title="E111">A296</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_2"><i><font color="Green" title="E120">A305</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_2"><i><font color="Green" title="E122">A307</font></i></a>, <a class="txt" href="jordan7.html#E38:27_3_1_2"><i><font color="Green" title="E123">A308</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_2"><i><font color="Green" title="E128">A313</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E12:27_3_1_2_3"/><i><font color="Green" title="E149">A351</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c16">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E40:27_3_1_2"><i><font color="Green" title="E125">A310</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E53:27_3_1_2"><i><font color="Green" title="E135">A320</font></i></a>, <a class="txt" href="jordan7.html#E54:27_3_1_2"><i><font color="Green" title="E136">A321</font></i></a>, <a class="txt" href="jordan7.html#E57:27_3_1_2"><i><font color="Green" title="E139">A324</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_2_3"><i><font color="Green" title="E144">A346</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2_3"><i><font color="Green" title="E145">A347</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E13:27_3_1_2_3"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E55:27_3_1_2"><i><font color="Green" title="E137">A322</font></i></a>, <a class="txt" href="jordan7.html#E56:27_3_1_2"><i><font color="Green" title="E138">A323</font></i></a>, <a class="txt" href="jordan7.html#E58:27_3_1_2"><i><font color="Green" title="E140">A325</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_2_3"><i><font color="Green" title="E142">A344</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_2_3"><i><font color="Green" title="E144">A346</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_2_3"><i><font color="Green" title="E145">A347</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E14:27_3_1_2_3"/><b>then </b>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E47:27_3_1_2"><i><font color="Green" title="E131">A316</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_2"><i><font color="Green" title="E132">A317</font></i></a>, <a class="txt" href="jordan7.html#E51:27_3_1_2"><i><font color="Green" title="E133">A318</font></i></a>, <a class="txt" href="jordan7.html#E52:27_3_1_2"><i><font color="Green" title="E134">A319</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_2_3"><i><font color="Green" title="E143">A345</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_2_3"><i><font color="Green" title="E147">A349</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_2_3"><i><font color="Green" title="E148">A350</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_2_3"><i><font color="Green" title="E149">A351</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E15:27_3_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E16:27_3_1_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_3_1_2_3"><i><font color="Green" title="E148">A350</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E62:27_3_1_2"/><b>then </b>
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E60:27_3_1_2"><i><font color="Green" title="E141">A326</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E63:27_3_1_2"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_2"><i><font color="Green" title="E116">A301</font></i></a>, <a class="txt" href="jordan7.html#E36:27_3_1_2"><i><font color="Green" title="E121">A306</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_3"><b>suppose </b></a><a NAME="E1:27_3_1_3"/><i><font color="Green" title="E100">A352</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/><div class="add"><a NAME="E2:27_3_1_3"/><b>then </b><i><font color="Green" title="E101">A353</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_3_1_3"/><i><font color="Green" title="E102">A354</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:27_3_1_3"/><i><font color="Green" title="E103">A355</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_3_1_3"/><i><font color="Green" title="E104">A356</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3"><i><font color="Green" title="E99">A209</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_3_1_3"/><i><font color="Green" title="E105">A357</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_3_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3"><i><font color="Green" title="E99">A209</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E8:27_3_1_3"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:27_3_1_3"/><b>then </b><i><font color="Green" title="E106">A358</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E10:27_3_1_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E11:27_3_1_3"/><b>then </b><i><font color="Green" title="E107">A359</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:27_3_1_3"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_3_1_3"/><b>then </b><i><font color="Green" title="E108">A360</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_3"><i><font color="Green" title="E102">A354</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_3_1_3"/><b>then </b><i><font color="Green" title="E109">A361</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_3_1_3"/><i><font color="Green" title="E110">A362</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_3"><i><font color="Green" title="E107">A359</font></i></a>, <a class="txt" href="jordan7.html#E13:27_3_1_3"><i><font color="Green" title="E108">A360</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E16:27_3_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_3_1_3"/><b>then </b><i><font color="Green" title="E111">A363</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_3_1_3"><i><font color="Green" title="E109">A361</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E19:27_3_1_3"/><i><font color="Green" title="E112">A364</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_3"><i><font color="Green" title="E110">A362</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c12">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E20:27_3_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E21:27_3_1_3"/><b>then </b><i><font color="Green" title="E113">A365</font></i>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E22:27_3_1_3"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_3_1_3"/><b>then </b><i><font color="Green" title="E114">A366</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E115">A367</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_3"><i><font color="Green" title="E110">A362</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E25:27_3_1_3"/><b>then </b><i><font color="Green" title="E116">A368</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_3"><i><font color="Green" title="E114">A366</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><i><font color="Green" title="E117">A369</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E19:27_3_1_3"><i><font color="Green" title="E112">A364</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_3"><i><font color="Green" title="E104">A356</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_3"><i><font color="Green" title="E106">A358</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E27:27_3_1_3"/><i><font color="Green" title="E118">A370</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_3"><i><font color="Green" title="E110">A362</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:27_3_1_3"/><b>then </b><i><font color="Green" title="E119">A371</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E29:27_3_1_3"/><i><font color="Green" title="E120">A372</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_3"><i><font color="Green" title="E101">A353</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3"><i><font color="Green" title="E113">A365</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E30:27_3_1_3"/><i><font color="Green" title="E121">A373</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27_3_1_3"><i><font color="Green" title="E118">A370</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_3_1_3"/><i><font color="Green" title="E122">A374</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_3"><i><font color="Green" title="E110">A362</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E32:27_3_1_3"/><i><font color="Green" title="E123">A375</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_3"><i><font color="Green" title="E103">A355</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_3"><i><font color="Green" title="E104">A356</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_3"><i><font color="Green" title="E105">A357</font></i></a>, <a class="txt" href="jordan7.html#E26:27_3_1_3"><i><font color="Green" title="E117">A369</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E33:27_3_1_3"/><i><font color="Green" title="E124">A376</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_3_1_3"><i><font color="Green" title="E108">A360</font></i></a>, <a class="txt" href="jordan7.html#E24:27_3_1_3"><i><font color="Green" title="E115">A367</font></i></a>, <a class="txt" href="jordan7.html#E31:27_3_1_3"><i><font color="Green" title="E122">A374</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:27_3_1_3"/><b>then </b><i><font color="Green" title="E125">A377</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E35:27_3_1_3"/><i><font color="Green" title="E126">A378</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E36:27_3_1_3"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:27_3_1_3"/><b>then </b><i><font color="Green" title="E127">A379</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E38:27_3_1_3"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_3"><i><font color="Green" title="E97">A207</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3"><i><font color="Green" title="E98">A208</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E39:27_3_1_3"/><b>then </b><i><font color="Green" title="E128">A380</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E40:27_3_1_3"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E30:27_3_1_3"><i><font color="Green" title="E121">A373</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:27_3_1_3"/><b>then </b><i><font color="Green" title="E129">A381</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_3"><i><font color="Green" title="E101">A353</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3"><i><font color="Green" title="E113">A365</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E42:27_3_1_3"/><i><font color="Green" title="E130">A382</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E34:27_3_1_3"><i><font color="Green" title="E125">A377</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_3"><i><font color="Green" title="E126">A378</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E43:27_3_1_3"/><i><font color="Green" title="E131">A383</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_3_1_3"><i><font color="Green" title="E110">A362</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_3"><i><font color="Green" title="E111">A363</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E44:27_3_1_3"/><i><font color="Green" title="E132">A384</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E45:27_3_1_3"/><i><font color="Green" title="E133">A385</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_3"><i><font color="Green" title="E126">A378</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E46:27_3_1_3"/><i><font color="Green" title="E134">A386</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27_3_1_3"><i><font color="Green" title="E131">A383</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><a NAME="E47:27_3_1_3"/><i><font color="Green" title="E135">A387</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_3_1_3"><i><font color="Green" title="E108">A360</font></i></a>, <a class="txt" href="jordan7.html#E24:27_3_1_3"><i><font color="Green" title="E115">A367</font></i></a>, <a class="txt" href="jordan7.html#E31:27_3_1_3"><i><font color="Green" title="E122">A374</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E30:27_3_1_3"><i><font color="Green" title="E121">A373</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_3"><i><font color="Green" title="E126">A378</font></i></a>, <a class="txt" href="jordan7.html#E46:27_3_1_3"><i><font color="Green" title="E134">A386</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/><a NAME="E49:27_3_1_3"/><i><font color="Green" title="E136">A388</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_3_3"/><i><font color="Green" title="E137">A389</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:27_3_1_3_3"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E25:27_3_1_3"><i><font color="Green" title="E116">A368</font></i></a>, <a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_3_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_3_1_3_3"><i><font color="Green" title="E137">A389</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_3_3"/><i><font color="Green" title="E138">A390</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_3_3"/><i><font color="Green" title="E139">A391</font></i>: 
( ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_3_3"><i><font color="Green" title="E138">A390</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_3"><i><font color="Green" title="E114">A366</font></i></a>, <a class="txt" href="jordan7.html#E24:27_3_1_3"><i><font color="Green" title="E115">A367</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E8:27_3_1_3_3"/><b>then </b><i><font color="Green" title="E140">A392</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E9:27_3_1_3_3"/><i><font color="Green" title="E141">A393</font></i>: 
not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E31:27_3_1_3"><i><font color="Green" title="E122">A374</font></i></a>, <a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:27_3_1_3_3"/><i><font color="Green" title="E142">A394</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E34:27_3_1_3"><i><font color="Green" title="E125">A377</font></i></a>, <a class="txt" href="jordan7.html#E35:27_3_1_3"><i><font color="Green" title="E126">A378</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_1"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_3_1_3_3"><i><font color="Green" title="E142">A394</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_3_3"><i><font color="Green" title="E141">A393</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_3_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E25:27_3_1_3"><i><font color="Green" title="E116">A368</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E12:27_3_1_3_3"/><b>then </b><i><font color="Green" title="E143">A395</font></i>: 
( ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) or ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_3_3"><i><font color="Green" title="E138">A390</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E13:27_3_1_3_3"/><i><font color="Green" title="E144">A396</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E27:27_3_1_3"><i><font color="Green" title="E118">A370</font></i></a>, <a class="txt" href="jordan7.html#E28:27_3_1_3"><i><font color="Green" title="E119">A371</font></i></a>, <a class="txt" href="jordan7.html#E29:27_3_1_3"><i><font color="Green" title="E120">A372</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_3"><i><font color="Green" title="E127">A379</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_3_3"><i><font color="Green" title="E140">A392</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E14:27_3_1_3_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E15:27_3_1_3_3"/><b>then </b><i><font color="Green" title="E145">A397</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green" title="E59">A36</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_3"><i><font color="Green" title="E101">A353</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E146">A398</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_2"/><i><font color="Green" title="E147">A399</font></i>: 
not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_3_3_2"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_3_3"><i><font color="Green" title="E139">A391</font></i></a>, <a class="txt" href="jordan7.html#E15:27_3_1_3_3"><i><font color="Green" title="E145">A397</font></i></a>, <a class="ref" href="jordan6.html#T70" title="JORDAN6:th.70">JORDAN6:70</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_3_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3_3_2"><i><font color="Green" title="E147">A399</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E147">A400</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_3"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span><br/><a NAME="E2:27_3_1_3_3_3"/><b>then </b>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_3_1_3_3"><i><font color="Green" title="E146">A398</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_3_3_3"/><b>then </b><i><font color="Green" title="E148">A401</font></i>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_3_3_3"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E6:27_3_1_3_3"><i><font color="Green" title="E139">A391</font></i></a>, <a class="txt" href="jordan7.html#E13:27_3_1_3_3"><i><font color="Green" title="E144">A396</font></i></a>, <a class="ref" href="jordan6.html#T69" title="JORDAN6:th.69">JORDAN6:69</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_3_3_3"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_3_3_3"><i><font color="Green" title="E148">A401</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E18:27_3_1_3_3"/><b>then </b><i><font color="Green" title="E148">A402</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E12:27_3_1_3_3"><i><font color="Green" title="E143">A395</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/>
<a NAME="E19:27_3_1_3_3"/>
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E5:27_3_1_3_3"><i><font color="Green" title="E138">A390</font></i></a>, <a class="txt" href="jordan7.html#E17:27_3_1_3_3"><i><font color="Green" title="E147">A400</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:27_3_1_3_3"/><i><font color="Green" title="E149">A403</font></i>: 
( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">rz</font> = <font color="Maroon" title="c16">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3_3"><i><font color="Green" title="E149">A403</font></i></a></i>;<br/>
<a NAME="E23:27_3_1_3_3"/><i><font color="Green" title="E150">A404</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27_3_1_3"><i><font color="Green" title="E108">A360</font></i></a>, <a class="txt" href="jordan7.html#E24:27_3_1_3"><i><font color="Green" title="E115">A367</font></i></a>, <a class="txt" href="jordan7.html#E31:27_3_1_3"><i><font color="Green" title="E122">A374</font></i></a>, <a class="txt" href="jordan7.html#E34:27_3_1_3"><i><font color="Green" title="E125">A377</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E24:27_3_1_3_3"/>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:27_3_1_3_3"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_3"><i><font color="Green" title="E135">A387</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3_3"><i><font color="Green" title="E149">A403</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:27_3_1_3_3"/><b>then </b>
( <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E1:27_3_1_3"><i><font color="Green" title="E100">A352</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3"><i><font color="Green" title="E113">A365</font></i></a>, <a class="txt" href="jordan7.html#E46:27_3_1_3"><i><font color="Green" title="E134">A386</font></i></a>, <a class="txt" href="jordan7.html#E18:27_3_1_3_3"><i><font color="Green" title="E148">A402</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3_3"><i><font color="Green" title="E149">A403</font></i></a>, <a class="txt" href="jordan7.html#E23:27_3_1_3_3"><i><font color="Green" title="E150">A404</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<a NAME="E27:27_3_1_3_3"/><b>then </b>
<font color="Maroon" title="c17">rz</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E28:27_3_1_3_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_3_1_3_3"><i><font color="Green" title="E138">A390</font></i></a>, <a class="txt" href="jordan7.html#E21:27_3_1_3_3"><i><font color="Green" title="E149">A403</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E50:27_3_1_3"/>
<font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_3_1_3_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_3_4"/><i><font color="Green" title="E137">A405</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_3_4"/><i><font color="Green" title="E138">A406</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">p1</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">sy</font> = <font color="Maroon" title="c15">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_3_4"/><i><font color="Green" title="E139">A407</font></i>: 
( <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_3_1_3_4"/><i><font color="Green" title="E140">A408</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_3_4"/><i><font color="Green" title="E141">A409</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E25:27_3_1_3"><i><font color="Green" title="E116">A368</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green" title="E142">A410</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_4_1"/>
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_3_1_3_4_1"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">sy</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green" title="E55">A32</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_3_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E45:27_3_1_3"><i><font color="Green" title="E133">A385</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_3"><i><font color="Green" title="E135">A387</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_3_4"><i><font color="Green" title="E139">A407</font></i></a>, <a class="txt" href="jordan7.html#E9:27_3_1_3_4"><i><font color="Green" title="E141">A409</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_3_4"/><i><font color="Green" title="E143">A411</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E25:27_3_1_3"><i><font color="Green" title="E116">A368</font></i></a>, <a class="txt" href="jordan7.html#E32:27_3_1_3"><i><font color="Green" title="E123">A375</font></i></a>, <a class="txt" href="jordan7.html#E33:27_3_1_3"><i><font color="Green" title="E124">A376</font></i></a>, <a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E12:27_3_1_3_4"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c16">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E2:27_3_1_3"><i><font color="Green" title="E101">A353</font></i></a>, <a class="txt" href="jordan7.html#E37:27_3_1_3"><i><font color="Green" title="E127">A379</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_3_4"><i><font color="Green" title="E140">A408</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E13:27_3_1_3_4"/><b>then </b><i><font color="Green" title="E144">A412</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E37:27_3_1_3"><i><font color="Green" title="E127">A379</font></i></a>, <a class="txt" href="jordan7.html#E41:27_3_1_3"><i><font color="Green" title="E129">A381</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_3_4"><i><font color="Green" title="E138">A406</font></i></a>, <a class="txt" href="jordan7.html#E10:27_3_1_3_4"><i><font color="Green" title="E142">A410</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E14:27_3_1_3_4"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E44:27_3_1_3"><i><font color="Green" title="E132">A384</font></i></a>, <a class="txt" href="jordan7.html#E45:27_3_1_3"><i><font color="Green" title="E133">A385</font></i></a>, <a class="txt" href="jordan7.html#E47:27_3_1_3"><i><font color="Green" title="E135">A387</font></i></a>, <a class="txt" href="jordan7.html#E3:27_3_1_3_4"><i><font color="Green" title="E137">A405</font></i></a>, <a class="txt" href="jordan7.html#E7:27_3_1_3_4"><i><font color="Green" title="E139">A407</font></i></a>, <a class="txt" href="jordan7.html#E8:27_3_1_3_4"><i><font color="Green" title="E140">A408</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<a NAME="E15:27_3_1_3_4"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E39:27_3_1_3"><i><font color="Green" title="E128">A380</font></i></a>, <a class="txt" href="jordan7.html#E42:27_3_1_3"><i><font color="Green" title="E130">A382</font></i></a>, <a class="txt" href="jordan7.html#E4:27_3_1_3_4"><i><font color="Green" title="E138">A406</font></i></a>, <a class="txt" href="jordan7.html#E11:27_3_1_3_4"><i><font color="Green" title="E143">A411</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:27_3_1_3_4"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27_3_1_3_4"><i><font color="Green" title="E144">A412</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:27_3_1_3_4"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_3_1_3_4"><i><font color="Green" title="E143">A411</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E51:27_3_1_3"/><b>then </b>
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_3"><i><font color="Green" title="E96">A206</font></i></a>, <a class="txt" href="jordan7.html#E49:27_3_1_3"><i><font color="Green" title="E136">A388</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E52:27_3_1_3"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c11">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E43:27_3_1_3"><i><font color="Green" title="E131">A383</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E97:27"/>
for <font color="Olive" title="b1">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> holds <br/> <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Olive" title="b1">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> implies  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c10">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font> )</span><br/>

<b>assume </b><a NAME="E1:27_4"/><i><font color="Green" title="E96">A413</font></i>: 
<font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c10">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font></span><br/>

<b>set </b><font color="Maroon" title="c11">i</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>;<br/>
<a NAME="E2:27_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E3:27_4"/><b>then </b><i><font color="Green" title="E97">A414</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E4:27_4"/><i><font color="Green" title="E98">A415</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E5:27_4"/><i><font color="Green" title="E99">A416</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E6:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E63:27"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_4"/><b>then </b><i><font color="Green" title="E100">A417</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_4"><i><font color="Green" title="E97">A414</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E5:27_4"><i><font color="Green" title="E99">A416</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E8:27_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E9:27_4"/><b>then </b><i><font color="Green" title="E101">A418</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:27_4"/><b>then </b><i><font color="Green" title="E102">A419</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E11:27_4"/><i><font color="Green" title="E103">A420</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E9:27_4"><i><font color="Green" title="E101">A418</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E12:27_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E13:27_4"/><b>then </b><i><font color="Green" title="E104">A421</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:27_4"/><i><font color="Green" title="E105">A422</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E10:27_4"><i><font color="Green" title="E102">A419</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E15:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:27_4"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/>
<a NAME="E17:27_4"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E18:27_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E19:27_4"/><b>then </b><i><font color="Green" title="E106">A423</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E14:27_4"><i><font color="Green" title="E105">A422</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:27_4"/><i><font color="Green" title="E107">A424</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E11:27_4"><i><font color="Green" title="E103">A420</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:27_4"/><b>then </b><i><font color="Green" title="E108">A425</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E7:27_4"><i><font color="Green" title="E100">A417</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E22:27_4"/><i><font color="Green" title="E109">A426</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27_4"><i><font color="Green" title="E107">A424</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E23:27_4"/><b>then </b><i><font color="Green" title="E110">A427</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E24:27_4"/><i><font color="Green" title="E111">A428</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E10:27_4"><i><font color="Green" title="E102">A419</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E25:27_4"/><i><font color="Green" title="E112">A429</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E21:27_4"><i><font color="Green" title="E108">A425</font></i></a>, <a class="txt" href="jordan7.html#E22:27_4"><i><font color="Green" title="E109">A426</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><i><font color="Green" title="E113">A430</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_4_1"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_4_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27_4"><i><font color="Green" title="E112">A429</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_4_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_4_1"/><b>then </b>
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E7:27_4"><i><font color="Green" title="E100">A417</font></i></a>, <a class="txt" href="jordan7.html#E13:27_4"><i><font color="Green" title="E104">A421</font></i></a>, <a class="txt" href="jordan7.html#E19:27_4"><i><font color="Green" title="E106">A423</font></i></a>, <a class="txt" href="jordan7.html#E20:27_4"><i><font color="Green" title="E107">A424</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E27:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E24:27_4"><i><font color="Green" title="E111">A428</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:27_4"/><b>then </b><i><font color="Green" title="E114">A431</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:27_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green" title="E76">A53</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E30:27_4"/><b>then </b><i><font color="Green" title="E115">A432</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E31:27_4"/><i><font color="Green" title="E116">A433</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E22:27_4"><i><font color="Green" title="E109">A426</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E32:27_4"/><i><font color="Green" title="E117">A434</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E22:27_4"><i><font color="Green" title="E109">A426</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E33:27_4"/><i><font color="Green" title="E118">A435</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E10:27_4"><i><font color="Green" title="E102">A419</font></i></a>, <a class="txt" href="jordan7.html#E13:27_4"><i><font color="Green" title="E104">A421</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E34:27_4"/><i><font color="Green" title="E119">A436</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E24:27_4"><i><font color="Green" title="E111">A428</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E22:27_4"><i><font color="Green" title="E109">A426</font></i></a>, <a class="txt" href="jordan7.html#E28:27_4"><i><font color="Green" title="E114">A431</font></i></a>, <a class="txt" href="jordan7.html#E33:27_4"><i><font color="Green" title="E118">A435</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E36:27_4"/><i><font color="Green" title="E120">A437</font></i>: 
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:27_4_2"/><i><font color="Green" title="E121">A438</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/>

<a NAME="E2:27_4_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E3:27_4_2"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_4_2"><i><font color="Green" title="E121">A438</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_4_2"/><i><font color="Green" title="E122">A439</font></i>: 
( <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp; (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) ) )
 ;<br/>
<a NAME="E6:27_4_2"/><i><font color="Green" title="E123">A440</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:27_4_2"/><i><font color="Green" title="E124">A441</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E10:27_4"><i><font color="Green" title="E102">A419</font></i></a>, <a class="txt" href="jordan7.html#E13:27_4"><i><font color="Green" title="E104">A421</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:27_4_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E24:27_4"><i><font color="Green" title="E111">A428</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:27_4_2"/><b>then </b>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E6:27_4_2"><i><font color="Green" title="E123">A440</font></i></a>, <a class="txt" href="jordan7.html#E7:27_4_2"><i><font color="Green" title="E124">A441</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E10:27_4_2"/><b>then </b><i><font color="Green" title="E125">A442</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27_4"><i><font color="Green" title="E118">A435</font></i></a>, <a class="txt" href="jordan7.html#E34:27_4"><i><font color="Green" title="E119">A436</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E11:27_4_2"/><i><font color="Green" title="E126">A443</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_2_1_1"/>
( (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c1">P</font> &amp; ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or  not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> ) ) or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E5:27_4_2"><i><font color="Green" title="E122">A439</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:27_4_2_1_1_1"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c15">p</font>,<font color="Maroon" title="c1">P</font> &amp; ( <font color="Maroon" title="c15">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or  not <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> )</span><br/><div class="add"><a NAME="E2:27_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E127">A444</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c15">p</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27_4"><i><font color="Green" title="E113">A430</font></i></a>, <a class="txt" href="jordan7.html#E30:27_4"><i><font color="Green" title="E115">A432</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_4_2_1_1_1"/><i><font color="Green" title="E128">A445</font></i>: 
( <font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">rz</font> = <font color="Maroon" title="c16">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4_2_1_1_1"><i><font color="Green" title="E128">A445</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c18">z</font> = <font color="Maroon" title="c16">z</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> )</span><br/><a NAME="E6:27_4_2_1_1_1"/><i><font color="Green" title="E129">A446</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font> &amp; <font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_4_2_1_1_1"><i><font color="Green" title="E128">A445</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E7:27_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E130">A447</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">rz</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E21:27_4"><i><font color="Green" title="E108">A425</font></i></a>, <a class="txt" href="jordan7.html#E23:27_4"><i><font color="Green" title="E110">A427</font></i></a>, <a class="txt" href="jordan7.html#E30:27_4"><i><font color="Green" title="E115">A432</font></i></a>, <a class="txt" href="jordan7.html#E33:27_4"><i><font color="Green" title="E118">A435</font></i></a>, <a class="txt" href="jordan7.html#E2:27_4_2_1_1_1"><i><font color="Green" title="E127">A444</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4_2_1_1_1"><i><font color="Green" title="E128">A445</font></i></a>, <a class="ref" href="jordan7.html#T20" target="_self" title="JORDAN7:th.20">Th20</a></i>;<br/><b>thus </b><a NAME="E8:27_4_2_1_1_1"/>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_4_2_1_1_1"><i><font color="Green" title="E128">A445</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> )</span><br/><a NAME="E9:27_4_2_1_1_1"/>
<font color="Maroon" title="c17">rz</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E24:27_4"><i><font color="Green" title="E111">A428</font></i></a>, <a class="txt" href="jordan7.html#E6:27_4_2_1_1_1"><i><font color="Green" title="E129">A446</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>;<br/><b>hence </b><a NAME="E10:27_4_2_1_1_1"/>
( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_4_2"><i><font color="Green" title="E122">A439</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4_2_1_1_1"><i><font color="Green" title="E128">A445</font></i></a>, <a class="txt" href="jordan7.html#E7:27_4_2_1_1_1"><i><font color="Green" title="E130">A447</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:27_4_2_1_1_2"/>
( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:27_4_2_1_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green" title="E55">A32</font></i></a>, <a class="txt" href="jordan7.html#E5:27_4_2"><i><font color="Green" title="E122">A439</font></i></a>, <a class="txt" href="jordan7.html#E10:27_4_2"><i><font color="Green" title="E125">A442</font></i></a>, <a class="txt" href="jordan7.html#E11:27_4_2"><i><font color="Green" title="E126">A443</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:27_4_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E37:27_4"/>
<font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_4_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_4_3"/><i><font color="Green" title="E121">A448</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">g2</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_4_3"/><i><font color="Green" title="E122">A449</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E3:27_4_3"><i><font color="Green" title="E121">A448</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">p1</font> = <font color="Maroon" title="c14">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">sy</font> = <font color="Maroon" title="c15">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E3:27_4_3"><i><font color="Green" title="E121">A448</font></i></a></i>;<br/>
<a NAME="E7:27_4_3"/><i><font color="Green" title="E123">A450</font></i>: 
( <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_4_3"><i><font color="Green" title="E121">A448</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_4_3"/><i><font color="Green" title="E124">A451</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">sy</font> &amp; <font color="Maroon" title="c17">sy</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_4_3"><i><font color="Green" title="E121">A448</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_4_3"/><i><font color="Green" title="E125">A452</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E10:27_4_3"/><i><font color="Green" title="E126">A453</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E11:27_4_3"/><i><font color="Green" title="E127">A454</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c16">p1</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E21:27_4"><i><font color="Green" title="E108">A425</font></i></a>, <a class="txt" href="jordan7.html#E30:27_4"><i><font color="Green" title="E115">A432</font></i></a>, <a class="txt" href="jordan7.html#E31:27_4"><i><font color="Green" title="E116">A433</font></i></a>, <a class="txt" href="jordan7.html#E32:27_4"><i><font color="Green" title="E117">A434</font></i></a>, <a class="txt" href="jordan7.html#E33:27_4"><i><font color="Green" title="E118">A435</font></i></a>, <a class="txt" href="jordan7.html#E3:27_4_3"><i><font color="Green" title="E121">A448</font></i></a>, <a class="txt" href="jordan7.html#E7:27_4_3"><i><font color="Green" title="E123">A450</font></i></a>, <a class="txt" href="jordan7.html#E8:27_4_3"><i><font color="Green" title="E124">A451</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_3_1_1"/>
( <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c16">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1_1"><b>case </b></a><a NAME="E1:27_4_3_1_1_1"/>
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )</span><br/><div class="add"><b>hence </b><a NAME="E2:27_4_3_1_1_1"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27_4"><i><font color="Green" title="E112">A429</font></i></a>, <a class="txt" href="jordan7.html#E30:27_4"><i><font color="Green" title="E115">A432</font></i></a>, <a class="txt" href="jordan7.html#E10:27_4_3"><i><font color="Green" title="E126">A453</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1_2"><b>case </b></a><a NAME="E1:27_4_3_1_1_2"/>
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )</span><br/><div class="add"><b>hence </b><a NAME="E2:27_4_3_1_1_2"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c16">p1</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27_4"><i><font color="Green" title="E112">A429</font></i></a>, <a class="txt" href="jordan7.html#E30:27_4"><i><font color="Green" title="E115">A432</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4_3"><i><font color="Green" title="E122">A449</font></i></a>, <a class="txt" href="jordan7.html#E11:27_4_3"><i><font color="Green" title="E127">A454</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:27_4_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font> or ( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> ) ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:27_4_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_4_3"><i><font color="Green" title="E125">A452</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self" title="JORDAN7:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E38:27_4"/><b>then </b>
<font color="Maroon" title="c10">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_4"><i><font color="Green" title="E96">A413</font></i></a>, <a class="txt" href="jordan7.html#E36:27_4"><i><font color="Green" title="E120">A437</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E39:27_4"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c10">W</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_4"><i><font color="Green" title="E98">A415</font></i></a>, <a class="txt" href="jordan7.html#E10:27_4"><i><font color="Green" title="E102">A419</font></i></a>, <a class="txt" href="jordan7.html#E13:27_4"><i><font color="Green" title="E104">A421</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E98:27"/><i><font color="Green" title="E96">A455</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:27_5"/><i><font color="Green" title="E97">A456</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E2:27_5"/><b>then </b><i><font color="Green" title="E98">A457</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:27_5"/><i><font color="Green" title="E99">A458</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:27_5"/><b>then </b><i><font color="Green" title="E100">A459</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:27_5"/><i><font color="Green" title="E101">A460</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_5_1"/>
( <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_1"><b>suppose </b></a><a NAME="E1:27_5_1_1"/><i><font color="Green" title="E102">A461</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_5_1_1"/><b>then </b><i><font color="Green" title="E103">A462</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_5_1_1"/><i><font color="Green" title="E104">A463</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_1"><i><font color="Green" title="E102">A461</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_5_1_1"/><i><font color="Green" title="E105">A464</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_1"><i><font color="Green" title="E102">A461</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_5_1_1"/><b>then </b><i><font color="Green" title="E106">A465</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_5_1_1"><i><font color="Green" title="E104">A463</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_5_1_1"/><i><font color="Green" title="E107">A466</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_5_1_1"><i><font color="Green" title="E105">A464</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_5_1_1"/><b>then </b><i><font color="Green" title="E108">A467</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_5_1_1"/><i><font color="Green" title="E109">A468</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:27_5_1_1"/><b>then </b><i><font color="Green" title="E110">A469</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_5_1_1"><i><font color="Green" title="E103">A462</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E10:27_5_1_1"/><i><font color="Green" title="E111">A470</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5_1_1"><i><font color="Green" title="E103">A462</font></i></a>, <a class="txt" href="jordan7.html#E8:27_5_1_1"><i><font color="Green" title="E109">A468</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:27_5_1_1"/><b>then </b><i><font color="Green" title="E112">A471</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_5_1_1"><i><font color="Green" title="E110">A469</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E12:27_5_1_1"/><i><font color="Green" title="E113">A472</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_5_1_1"><i><font color="Green" title="E111">A470</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:27_5_1_1"/><b>then </b><i><font color="Green" title="E114">A473</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_5_1_1"/><i><font color="Green" title="E115">A474</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5"><i><font color="Green" title="E101">A460</font></i></a>, <a class="txt" href="jordan7.html#E4:27_5_1_1"><i><font color="Green" title="E105">A464</font></i></a>, <a class="txt" href="jordan7.html#E10:27_5_1_1"><i><font color="Green" title="E111">A470</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E15:27_5_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_5_1_1"/><b>then </b><i><font color="Green" title="E116">A475</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E17:27_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="txt" href="jordan7.html#E8:27_5_1_1"><i><font color="Green" title="E109">A468</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_5_1_1"/><b>then </b><i><font color="Green" title="E117">A476</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E19:27_5_1_1"/>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E6:27_5_1_1"><i><font color="Green" title="E107">A466</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_5_1_1"/><b>then </b><i><font color="Green" title="E118">A477</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5_1_1"><i><font color="Green" title="E104">A463</font></i></a>, <a class="txt" href="jordan7.html#E4:27_5_1_1"><i><font color="Green" title="E105">A464</font></i></a>, <a class="txt" href="jordan7.html#E16:27_5_1_1"><i><font color="Green" title="E116">A475</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E21:27_5_1_1"/><i><font color="Green" title="E119">A478</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E11:27_5_1_1"><i><font color="Green" title="E112">A471</font></i></a>, <a class="txt" href="jordan7.html#E12:27_5_1_1"><i><font color="Green" title="E113">A472</font></i></a>, <a class="txt" href="jordan7.html#E18:27_5_1_1"><i><font color="Green" title="E117">A476</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:27_5_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5_1_1"><i><font color="Green" title="E106">A465</font></i></a>, <a class="txt" href="jordan7.html#E7:27_5_1_1"><i><font color="Green" title="E108">A467</font></i></a>, <a class="txt" href="jordan7.html#E11:27_5_1_1"><i><font color="Green" title="E112">A471</font></i></a>, <a class="txt" href="jordan7.html#E13:27_5_1_1"><i><font color="Green" title="E114">A473</font></i></a>, <a class="txt" href="jordan7.html#E14:27_5_1_1"><i><font color="Green" title="E115">A474</font></i></a>, <a class="txt" href="jordan7.html#E16:27_5_1_1"><i><font color="Green" title="E116">A475</font></i></a>, <a class="txt" href="jordan7.html#E18:27_5_1_1"><i><font color="Green" title="E117">A476</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E23:27_5_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27_5_1_1"><i><font color="Green" title="E118">A477</font></i></a>, <a class="txt" href="jordan7.html#E21:27_5_1_1"><i><font color="Green" title="E119">A478</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2"><b>suppose </b></a><a NAME="E1:27_5_1_2"/><i><font color="Green" title="E102">A479</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_5_1_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E1:27_5_1_2"><i><font color="Green" title="E102">A479</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_5_1_2_1_1"/><i><font color="Green" title="E103">A480</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E104">A481</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_5_1_2_1_1"/><i><font color="Green" title="E105">A482</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_1"><i><font color="Green" title="E103">A480</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_5_1_2_1_1"/><i><font color="Green" title="E106">A483</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E107">A484</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_5_1_2_1_1"><i><font color="Green" title="E105">A482</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_5_1_2_1_1"/><i><font color="Green" title="E108">A485</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_1"><i><font color="Green" title="E103">A480</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E7:27_5_1_2_1_1"/><i><font color="Green" title="E109">A486</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_5_1_2_1_1"><i><font color="Green" title="E106">A483</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5_1_2_1_1"><i><font color="Green" title="E104">A481</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E110">A487</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5_1_2_1_1"><i><font color="Green" title="E107">A484</font></i></a>, <a class="txt" href="jordan7.html#E6:27_5_1_2_1_1"><i><font color="Green" title="E108">A485</font></i></a>, <a class="txt" href="jordan7.html#E7:27_5_1_2_1_1"><i><font color="Green" title="E109">A486</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_5_1_2_1_1"/><i><font color="Green" title="E111">A488</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5_1_2_1_1"><i><font color="Green" title="E104">A481</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E112">A489</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E12:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_1"><i><font color="Green" title="E103">A480</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E113">A490</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E14:27_5_1_2_1_1"/><i><font color="Green" title="E114">A491</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_5_1_2_1_1"/><i><font color="Green" title="E115">A492</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_5_1_2_1_1"><i><font color="Green" title="E111">A488</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E116">A493</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E11:27_5_1_2_1_1"><i><font color="Green" title="E112">A489</font></i></a>, <a class="txt" href="jordan7.html#E13:27_5_1_2_1_1"><i><font color="Green" title="E113">A490</font></i></a>, <a class="txt" href="jordan7.html#E14:27_5_1_2_1_1"><i><font color="Green" title="E114">A491</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E17:27_5_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E13:27_5_1_2_1_1"><i><font color="Green" title="E113">A490</font></i></a>, <a class="txt" href="jordan7.html#E15:27_5_1_2_1_1"><i><font color="Green" title="E115">A492</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E117">A494</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E19:27_5_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E20:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E118">A495</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E119">A496</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E22:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E120">A497</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E6:27_5_1_2_1_1"><i><font color="Green" title="E108">A485</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E24:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E121">A498</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E20:27_5_1_2_1_1"><i><font color="Green" title="E118">A495</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E25:27_5_1_2_1_1"/><i><font color="Green" title="E122">A499</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27_5_1_2_1_1"><i><font color="Green" title="E119">A496</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E26:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E123">A500</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E124">A501</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E21:27_5_1_2_1_1"><i><font color="Green" title="E119">A496</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2

;<br/><a NAME="E29:27_5_1_2_1_1"/><i><font color="Green" title="E125">A502</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_5_1_2_1_1"><i><font color="Green" title="E108">A485</font></i></a>, <a class="txt" href="jordan7.html#E13:27_5_1_2_1_1"><i><font color="Green" title="E113">A490</font></i></a>, <a class="txt" href="jordan7.html#E18:27_5_1_2_1_1"><i><font color="Green" title="E117">A494</font></i></a>, <a class="txt" href="jordan7.html#E25:27_5_1_2_1_1"><i><font color="Green" title="E122">A499</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E30:27_5_1_2_1_1"/><i><font color="Green" title="E126">A503</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E24:27_5_1_2_1_1"><i><font color="Green" title="E121">A498</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E31:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E127">A504</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_5_1_2_1_1"><i><font color="Green" title="E110">A487</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E32:27_5_1_2_1_1"/><i><font color="Green" title="E128">A505</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E30:27_5_1_2_1_1"><i><font color="Green" title="E126">A503</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E33:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E129">A506</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E34:27_5_1_2_1_1"/><i><font color="Green" title="E130">A507</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E35:27_5_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27_5_1_2_1_1"><i><font color="Green" title="E119">A496</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E36:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E131">A508</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_5_1_2_1_1"><i><font color="Green" title="E120">A497</font></i></a>, <a class="txt" href="jordan7.html#E28:27_5_1_2_1_1"><i><font color="Green" title="E124">A501</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E132">A509</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_5_1_2_1_1"><i><font color="Green" title="E116">A493</font></i></a>, <a class="txt" href="jordan7.html#E29:27_5_1_2_1_1"><i><font color="Green" title="E125">A502</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E38:27_5_1_2_1_1"/><i><font color="Green" title="E133">A510</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27_5_1_2_1_1"><i><font color="Green" title="E131">A508</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E39:27_5_1_2_1_1"/><i><font color="Green" title="E134">A511</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E30:27_5_1_2_1_1"><i><font color="Green" title="E126">A503</font></i></a>, <a class="txt" href="jordan7.html#E34:27_5_1_2_1_1"><i><font color="Green" title="E130">A507</font></i></a>, <a class="txt" href="jordan7.html#E36:27_5_1_2_1_1"><i><font color="Green" title="E131">A508</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E40:27_5_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E41:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E135">A512</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E42:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="txt" href="jordan7.html#E4:27_5"><i><font color="Green" title="E100">A459</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E43:27_5_1_2_1_1"/><b>then </b><i><font color="Green" title="E136">A513</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E44:27_5_1_2_1_1"/><i><font color="Green" title="E137">A514</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E31:27_5_1_2_1_1"><i><font color="Green" title="E127">A504</font></i></a>, <a class="txt" href="jordan7.html#E32:27_5_1_2_1_1"><i><font color="Green" title="E128">A505</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E45:27_5_1_2_1_1"/><i><font color="Green" title="E138">A515</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E37:27_5_1_2_1_1"><i><font color="Green" title="E132">A509</font></i></a>, <a class="txt" href="jordan7.html#E38:27_5_1_2_1_1"><i><font color="Green" title="E133">A510</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E46:27_5_1_2_1_1"/><i><font color="Green" title="E139">A516</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E38:27_5_1_2_1_1"><i><font color="Green" title="E133">A510</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E47:27_5_1_2_1_1"/><i><font color="Green" title="E140">A517</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E6:27_5_1_2_1_1"><i><font color="Green" title="E108">A485</font></i></a>, <a class="txt" href="jordan7.html#E16:27_5_1_2_1_1"><i><font color="Green" title="E116">A493</font></i></a>, <a class="txt" href="jordan7.html#E27:27_5_1_2_1_1"><i><font color="Green" title="E123">A500</font></i></a>, <a class="txt" href="jordan7.html#E28:27_5_1_2_1_1"><i><font color="Green" title="E124">A501</font></i></a>, <a class="txt" href="jordan7.html#E29:27_5_1_2_1_1"><i><font color="Green" title="E125">A502</font></i></a>, <a class="txt" href="jordan7.html#E36:27_5_1_2_1_1"><i><font color="Green" title="E131">A508</font></i></a>, <a class="txt" href="jordan7.html#E43:27_5_1_2_1_1"><i><font color="Green" title="E136">A513</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E48:27_5_1_2_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E31:27_5_1_2_1_1"><i><font color="Green" title="E127">A504</font></i></a>, <a class="txt" href="jordan7.html#E33:27_5_1_2_1_1"><i><font color="Green" title="E129">A506</font></i></a>, <a class="txt" href="jordan7.html#E37:27_5_1_2_1_1"><i><font color="Green" title="E132">A509</font></i></a>, <a class="txt" href="jordan7.html#E39:27_5_1_2_1_1"><i><font color="Green" title="E134">A511</font></i></a>, <a class="txt" href="jordan7.html#E41:27_5_1_2_1_1"><i><font color="Green" title="E135">A512</font></i></a>, <a class="txt" href="jordan7.html#E43:27_5_1_2_1_1"><i><font color="Green" title="E136">A513</font></i></a>, <a class="txt" href="jordan7.html#E46:27_5_1_2_1_1"><i><font color="Green" title="E139">A516</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E49:27_5_1_2_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27_5_1_2_1_1"><i><font color="Green" title="E135">A512</font></i></a>, <a class="txt" href="jordan7.html#E43:27_5_1_2_1_1"><i><font color="Green" title="E136">A513</font></i></a>, <a class="txt" href="jordan7.html#E44:27_5_1_2_1_1"><i><font color="Green" title="E137">A514</font></i></a>, <a class="txt" href="jordan7.html#E45:27_5_1_2_1_1"><i><font color="Green" title="E138">A515</font></i></a>, <a class="txt" href="jordan7.html#E47:27_5_1_2_1_1"><i><font color="Green" title="E140">A517</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_5_1_2_1_2"/><i><font color="Green" title="E103">A518</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E104">A519</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_5_1_2_1_2"/><i><font color="Green" title="E105">A520</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:27_5_1_2_1_2"/><i><font color="Green" title="E106">A521</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_5_1_2_1_2"/><i><font color="Green" title="E107">A522</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_5_1_2_1_2"/><i><font color="Green" title="E108">A523</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5"><i><font color="Green" title="E101">A460</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E7:27_5_1_2_1_2"/><i><font color="Green" title="E109">A524</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E4:27_5_1_2_1_2"><i><font color="Green" title="E106">A521</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E110">A525</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5"><i><font color="Green" title="E101">A460</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E9:27_5_1_2_1_2"/><i><font color="Green" title="E111">A526</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E5:27_5_1_2_1_2"><i><font color="Green" title="E107">A522</font></i></a>, <a class="txt" href="jordan7.html#E6:27_5_1_2_1_2"><i><font color="Green" title="E108">A523</font></i></a>, <a class="txt" href="jordan7.html#E7:27_5_1_2_1_2"><i><font color="Green" title="E109">A524</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_5_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_5_1_2_1_2"><i><font color="Green" title="E110">A525</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E112">A527</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E12:27_5_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E13:27_5_1_2_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E113">A528</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E15:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E114">A529</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5_1_2_1_2"><i><font color="Green" title="E105">A520</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E17:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E115">A530</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E116">A531</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E14:27_5_1_2_1_2"><i><font color="Green" title="E113">A528</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E20:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E117">A532</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E18:27_5_1_2_1_2"><i><font color="Green" title="E115">A530</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><i><font color="Green" title="E118">A533</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E6:27_5_1_2_1_2"><i><font color="Green" title="E108">A523</font></i></a>, <a class="txt" href="jordan7.html#E11:27_5_1_2_1_2"><i><font color="Green" title="E112">A527</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E22:27_5_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_5_1_2_1_2"><i><font color="Green" title="E113">A528</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E119">A534</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_5_1_2_1_2"><i><font color="Green" title="E114">A529</font></i></a>, <a class="txt" href="jordan7.html#E19:27_5_1_2_1_2"><i><font color="Green" title="E116">A531</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E120">A535</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_5_1_2_1_2"><i><font color="Green" title="E111">A526</font></i></a>, <a class="txt" href="jordan7.html#E21:27_5_1_2_1_2"><i><font color="Green" title="E118">A533</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_5_1_2_1_2"/><i><font color="Green" title="E121">A536</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_5_1_2_1_2"><i><font color="Green" title="E119">A534</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E26:27_5_1_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5"><i><font color="Green" title="E97">A456</font></i></a>, <a class="txt" href="jordan7.html#E3:27_5"><i><font color="Green" title="E99">A458</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E122">A537</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E28:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_5"><i><font color="Green" title="E98">A457</font></i></a>, <a class="txt" href="jordan7.html#E4:27_5"><i><font color="Green" title="E100">A459</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E29:27_5_1_2_1_2"/><b>then </b><i><font color="Green" title="E123">A538</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E30:27_5_1_2_1_2"/><i><font color="Green" title="E124">A539</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E24:27_5_1_2_1_2"><i><font color="Green" title="E120">A535</font></i></a>, <a class="txt" href="jordan7.html#E25:27_5_1_2_1_2"><i><font color="Green" title="E121">A536</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_5_1_2_1_2"/><i><font color="Green" title="E125">A540</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E2:27_5_1_2_1_2"><i><font color="Green" title="E104">A519</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a></i>;<br/><a NAME="E32:27_5_1_2_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E9:27_5_1_2_1_2"><i><font color="Green" title="E111">A526</font></i></a>, <a class="txt" href="jordan7.html#E20:27_5_1_2_1_2"><i><font color="Green" title="E117">A532</font></i></a>, <a class="txt" href="jordan7.html#E21:27_5_1_2_1_2"><i><font color="Green" title="E118">A533</font></i></a>, <a class="txt" href="jordan7.html#E23:27_5_1_2_1_2"><i><font color="Green" title="E119">A534</font></i></a>, <a class="txt" href="jordan7.html#E29:27_5_1_2_1_2"><i><font color="Green" title="E123">A538</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E33:27_5_1_2_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_5_1_2_1_2"><i><font color="Green" title="E103">A518</font></i></a>, <a class="txt" href="jordan7.html#E27:27_5_1_2_1_2"><i><font color="Green" title="E122">A537</font></i></a>, <a class="txt" href="jordan7.html#E29:27_5_1_2_1_2"><i><font color="Green" title="E123">A538</font></i></a>, <a class="txt" href="jordan7.html#E30:27_5_1_2_1_2"><i><font color="Green" title="E124">A539</font></i></a>, <a class="txt" href="jordan7.html#E31:27_5_1_2_1_2"><i><font color="Green" title="E125">A540</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E99:27"/><i><font color="Green" title="E97">A541</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/>(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_6"/><i><font color="Green" title="E98">A542</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>and </b><br/><a NAME="E2:27_6"/><i><font color="Green" title="E99">A543</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/>

<a NAME="E3:27_6"/><i><font color="Green" title="E100">A544</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:27_6"/><i><font color="Green" title="E101">A545</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:27_6"/><i><font color="Green" title="E102">A546</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:27_6"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:27_6"/><b>then </b><i><font color="Green" title="E103">A547</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E8:27_6"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6"><i><font color="Green" title="E100">A544</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:27_6"/><b>then </b><i><font color="Green" title="E104">A548</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_6_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_1"><b>suppose </b></a><a NAME="E1:27_6_1_1"/><i><font color="Green" title="E105">A549</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:27_6_1_1"/><b>then </b><i><font color="Green" title="E106">A550</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_6_1_1"/><i><font color="Green" title="E107">A551</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_1"><i><font color="Green" title="E105">A549</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_6_1_1"/><i><font color="Green" title="E108">A552</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_1"><i><font color="Green" title="E105">A549</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_6_1_1"/><b>then </b><i><font color="Green" title="E109">A553</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_6_1_1"/><i><font color="Green" title="E110">A554</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6"><i><font color="Green" title="E100">A544</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6_1_1"><i><font color="Green" title="E106">A550</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_6_1_1"/><i><font color="Green" title="E111">A555</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_6"><i><font color="Green" title="E100">A544</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6_1_1"><i><font color="Green" title="E106">A550</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:27_6_1_1"/><b>then </b><i><font color="Green" title="E112">A556</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_6_1_1"><i><font color="Green" title="E110">A554</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E9:27_6_1_1"/><i><font color="Green" title="E113">A557</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27_6_1_1"><i><font color="Green" title="E111">A555</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:27_6_1_1"/><i><font color="Green" title="E114">A558</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6"><i><font color="Green" title="E102">A546</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6_1_1"><i><font color="Green" title="E108">A552</font></i></a>, <a class="txt" href="jordan7.html#E7:27_6_1_1"><i><font color="Green" title="E111">A555</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E11:27_6_1_1"/>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_1"><i><font color="Green" title="E109">A553</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:27_6_1_1"/><b>then </b><i><font color="Green" title="E115">A559</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6_1_1"><i><font color="Green" title="E107">A551</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6_1_1"><i><font color="Green" title="E108">A552</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E13:27_6_1_1"/><i><font color="Green" title="E116">A560</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E8:27_6_1_1"><i><font color="Green" title="E112">A556</font></i></a>, <a class="txt" href="jordan7.html#E9:27_6_1_1"><i><font color="Green" title="E113">A557</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:27_6_1_1"/><i><font color="Green" title="E117">A561</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27_6"><i><font color="Green" title="E103">A547</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6_1_1"><i><font color="Green" title="E107">A551</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6_1_1"><i><font color="Green" title="E108">A552</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E15:27_6_1_1"/><i><font color="Green" title="E118">A562</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_1"><i><font color="Green" title="E109">A553</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:27_6_1_1"/><i><font color="Green" title="E119">A563</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_1"><i><font color="Green" title="E109">A553</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E17:27_6_1_1"/><i><font color="Green" title="E120">A564</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E9:27_6_1_1"><i><font color="Green" title="E113">A557</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E18:27_6_1_1"/><i><font color="Green" title="E121">A565</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E19:27_6_1_1"/><i><font color="Green" title="E122">A566</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">i</font>
 ;<br/><a NAME="E20:27_6_1_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E9:27_6"><i><font color="Green" title="E104">A548</font></i></a>, <a class="txt" href="jordan7.html#E8:27_6_1_1"><i><font color="Green" title="E112">A556</font></i></a>, <a class="txt" href="jordan7.html#E10:27_6_1_1"><i><font color="Green" title="E114">A558</font></i></a>, <a class="txt" href="jordan7.html#E14:27_6_1_1"><i><font color="Green" title="E117">A561</font></i></a>, <a class="txt" href="jordan7.html#E15:27_6_1_1"><i><font color="Green" title="E118">A562</font></i></a>, <a class="txt" href="jordan7.html#E16:27_6_1_1"><i><font color="Green" title="E119">A563</font></i></a>, <a class="txt" href="jordan7.html#E17:27_6_1_1"><i><font color="Green" title="E120">A564</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E21:27_6_1_1"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E7:27_6"><i><font color="Green" title="E103">A547</font></i></a>, <a class="txt" href="jordan7.html#E9:27_6"><i><font color="Green" title="E104">A548</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6_1_1"><i><font color="Green" title="E107">A551</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6_1_1"><i><font color="Green" title="E108">A552</font></i></a>, <a class="txt" href="jordan7.html#E6:27_6_1_1"><i><font color="Green" title="E110">A554</font></i></a>, <a class="txt" href="jordan7.html#E7:27_6_1_1"><i><font color="Green" title="E111">A555</font></i></a>, <a class="txt" href="jordan7.html#E12:27_6_1_1"><i><font color="Green" title="E115">A559</font></i></a>, <a class="txt" href="jordan7.html#E13:27_6_1_1"><i><font color="Green" title="E116">A560</font></i></a>, <a class="txt" href="jordan7.html#E18:27_6_1_1"><i><font color="Green" title="E121">A565</font></i></a>, <a class="txt" href="jordan7.html#E19:27_6_1_1"><i><font color="Green" title="E122">A566</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2"><b>suppose </b></a><a NAME="E1:27_6_1_2"/><i><font color="Green" title="E105">A567</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_6_1_2_1"/>
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E1:27_6_1_2"><i><font color="Green" title="E105">A567</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_6_1_2_1_1"/><i><font color="Green" title="E106">A568</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E107">A569</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_6_1_2_1_1"/><i><font color="Green" title="E108">A570</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_1"><i><font color="Green" title="E106">A568</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E109">A571</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_6_1_2_1_1"/><i><font color="Green" title="E110">A572</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_1"><i><font color="Green" title="E106">A568</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_6_1_2_1_1"/><i><font color="Green" title="E111">A573</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_6_1_2_1_1"><i><font color="Green" title="E108">A570</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_6_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_6_1_2_1_1"><i><font color="Green" title="E107">A569</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E112">A574</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6_1_2_1_1"><i><font color="Green" title="E109">A571</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_2_1_1"><i><font color="Green" title="E110">A572</font></i></a>, <a class="txt" href="jordan7.html#E6:27_6_1_2_1_1"><i><font color="Green" title="E111">A573</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_6_1_2_1_1"/><i><font color="Green" title="E113">A575</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_6_1_2_1_1"><i><font color="Green" title="E107">A569</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E114">A576</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_6_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_1"><i><font color="Green" title="E106">A568</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E115">A577</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E13:27_6_1_2_1_1"/><i><font color="Green" title="E116">A578</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_6_1_2_1_1"/><i><font color="Green" title="E117">A579</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27_6_1_2_1_1"><i><font color="Green" title="E113">A575</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E118">A580</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E10:27_6_1_2_1_1"><i><font color="Green" title="E114">A576</font></i></a>, <a class="txt" href="jordan7.html#E12:27_6_1_2_1_1"><i><font color="Green" title="E115">A577</font></i></a>, <a class="txt" href="jordan7.html#E13:27_6_1_2_1_1"><i><font color="Green" title="E116">A578</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_6_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_6_1_2_1_1"><i><font color="Green" title="E115">A577</font></i></a>, <a class="txt" href="jordan7.html#E14:27_6_1_2_1_1"><i><font color="Green" title="E117">A579</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E119">A581</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E18:27_6_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E19:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E120">A582</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E121">A583</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E21:27_6_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E122">A584</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_2_1_1"><i><font color="Green" title="E110">A572</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E23:27_6_1_2_1_1"/><i><font color="Green" title="E123">A585</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E19:27_6_1_2_1_1"><i><font color="Green" title="E120">A582</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E24:27_6_1_2_1_1"/><i><font color="Green" title="E124">A586</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E22:27_6_1_2_1_1"><i><font color="Green" title="E122">A584</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E25:27_6_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E26:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E125">A587</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E126">A588</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E20:27_6_1_2_1_1"><i><font color="Green" title="E121">A583</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><i><font color="Green" title="E127">A589</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_6_1_2_1_1"><i><font color="Green" title="E110">A572</font></i></a>, <a class="txt" href="jordan7.html#E20:27_6_1_2_1_1"><i><font color="Green" title="E121">A583</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E12:27_6_1_2_1_1"><i><font color="Green" title="E115">A577</font></i></a>, <a class="txt" href="jordan7.html#E17:27_6_1_2_1_1"><i><font color="Green" title="E119">A581</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E29:27_6_1_2_1_1"/><i><font color="Green" title="E128">A590</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E23:27_6_1_2_1_1"><i><font color="Green" title="E123">A585</font></i></a>, <a class="txt" href="jordan7.html#E24:27_6_1_2_1_1"><i><font color="Green" title="E124">A586</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E129">A591</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_6_1_2_1_1"><i><font color="Green" title="E112">A574</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E31:27_6_1_2_1_1"/><i><font color="Green" title="E130">A592</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E29:27_6_1_2_1_1"><i><font color="Green" title="E128">A590</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_6_1_2_1_1"/><i><font color="Green" title="E131">A593</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E33:27_6_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_6_1_2_1_1"><i><font color="Green" title="E123">A585</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E34:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E132">A594</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_6_1_2_1_1"><i><font color="Green" title="E110">A572</font></i></a>, <a class="txt" href="jordan7.html#E26:27_6_1_2_1_1"><i><font color="Green" title="E125">A587</font></i></a>, <a class="txt" href="jordan7.html#E27:27_6_1_2_1_1"><i><font color="Green" title="E126">A588</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E35:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E133">A595</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_6_1_2_1_1"><i><font color="Green" title="E118">A580</font></i></a>, <a class="txt" href="jordan7.html#E28:27_6_1_2_1_1"><i><font color="Green" title="E127">A589</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E36:27_6_1_2_1_1"/><i><font color="Green" title="E134">A596</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27_6_1_2_1_1"><i><font color="Green" title="E132">A594</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E37:27_6_1_2_1_1"/><i><font color="Green" title="E135">A597</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E29:27_6_1_2_1_1"><i><font color="Green" title="E128">A590</font></i></a>, <a class="txt" href="jordan7.html#E32:27_6_1_2_1_1"><i><font color="Green" title="E131">A593</font></i></a>, <a class="txt" href="jordan7.html#E34:27_6_1_2_1_1"><i><font color="Green" title="E132">A594</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E38:27_6_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E39:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E136">A598</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E40:27_6_1_2_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6"><i><font color="Green" title="E100">A544</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E41:27_6_1_2_1_1"/><b>then </b><i><font color="Green" title="E137">A599</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E42:27_6_1_2_1_1"/><i><font color="Green" title="E138">A600</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E31:27_6_1_2_1_1"><i><font color="Green" title="E130">A592</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E43:27_6_1_2_1_1"/><i><font color="Green" title="E139">A601</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E31:27_6_1_2_1_1"><i><font color="Green" title="E130">A592</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E44:27_6_1_2_1_1"/><i><font color="Green" title="E140">A602</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E36:27_6_1_2_1_1"><i><font color="Green" title="E134">A596</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E45:27_6_1_2_1_1"/><i><font color="Green" title="E141">A603</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_2_1_1"><i><font color="Green" title="E110">A572</font></i></a>, <a class="txt" href="jordan7.html#E15:27_6_1_2_1_1"><i><font color="Green" title="E118">A580</font></i></a>, <a class="txt" href="jordan7.html#E26:27_6_1_2_1_1"><i><font color="Green" title="E125">A587</font></i></a>, <a class="txt" href="jordan7.html#E27:27_6_1_2_1_1"><i><font color="Green" title="E126">A588</font></i></a>, <a class="txt" href="jordan7.html#E28:27_6_1_2_1_1"><i><font color="Green" title="E127">A589</font></i></a>, <a class="txt" href="jordan7.html#E34:27_6_1_2_1_1"><i><font color="Green" title="E132">A594</font></i></a>, <a class="txt" href="jordan7.html#E41:27_6_1_2_1_1"><i><font color="Green" title="E137">A599</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E46:27_6_1_2_1_1"/><i><font color="Green" title="E142">A604</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E47:27_6_1_2_1_1"/>
( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E30:27_6_1_2_1_1"><i><font color="Green" title="E129">A591</font></i></a>, <a class="txt" href="jordan7.html#E31:27_6_1_2_1_1"><i><font color="Green" title="E130">A592</font></i></a>, <a class="txt" href="jordan7.html#E35:27_6_1_2_1_1"><i><font color="Green" title="E133">A595</font></i></a>, <a class="txt" href="jordan7.html#E36:27_6_1_2_1_1"><i><font color="Green" title="E134">A596</font></i></a>, <a class="txt" href="jordan7.html#E37:27_6_1_2_1_1"><i><font color="Green" title="E135">A597</font></i></a>, <a class="txt" href="jordan7.html#E39:27_6_1_2_1_1"><i><font color="Green" title="E136">A598</font></i></a>, <a class="txt" href="jordan7.html#E41:27_6_1_2_1_1"><i><font color="Green" title="E137">A599</font></i></a>, <a class="txt" href="jordan7.html#E42:27_6_1_2_1_1"><i><font color="Green" title="E138">A600</font></i></a>, <a class="txt" href="jordan7.html#E43:27_6_1_2_1_1"><i><font color="Green" title="E139">A601</font></i></a>, <a class="txt" href="jordan7.html#E44:27_6_1_2_1_1"><i><font color="Green" title="E140">A602</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E48:27_6_1_2_1_1"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_6_1_2_1_1"><i><font color="Green" title="E112">A574</font></i></a>, <a class="txt" href="jordan7.html#E15:27_6_1_2_1_1"><i><font color="Green" title="E118">A580</font></i></a>, <a class="txt" href="jordan7.html#E28:27_6_1_2_1_1"><i><font color="Green" title="E127">A589</font></i></a>, <a class="txt" href="jordan7.html#E29:27_6_1_2_1_1"><i><font color="Green" title="E128">A590</font></i></a>, <a class="txt" href="jordan7.html#E34:27_6_1_2_1_1"><i><font color="Green" title="E132">A594</font></i></a>, <a class="txt" href="jordan7.html#E39:27_6_1_2_1_1"><i><font color="Green" title="E136">A598</font></i></a>, <a class="txt" href="jordan7.html#E41:27_6_1_2_1_1"><i><font color="Green" title="E137">A599</font></i></a>, <a class="txt" href="jordan7.html#E45:27_6_1_2_1_1"><i><font color="Green" title="E141">A603</font></i></a>, <a class="txt" href="jordan7.html#E46:27_6_1_2_1_1"><i><font color="Green" title="E142">A604</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_6_1_2_1_2"/><i><font color="Green" title="E106">A605</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><div class="add"><a NAME="E2:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E107">A606</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_6_1_2_1_2"/><i><font color="Green" title="E108">A607</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:27_6_1_2_1_2"/><i><font color="Green" title="E109">A608</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_6_1_2_1_2"/><i><font color="Green" title="E110">A609</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_6_1_2_1_2"/><i><font color="Green" title="E111">A610</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6"><i><font color="Green" title="E102">A546</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E7:27_6_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_6_1_2_1_2"><i><font color="Green" title="E109">A608</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E112">A611</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="txt" href="jordan7.html#E5:27_6_1_2_1_2"><i><font color="Green" title="E110">A609</font></i></a>, <a class="txt" href="jordan7.html#E6:27_6_1_2_1_2"><i><font color="Green" title="E111">A610</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_6_1_2_1_2"/><i><font color="Green" title="E113">A612</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="txt" href="jordan7.html#E6:27_6_1_2_1_2"><i><font color="Green" title="E111">A610</font></i></a></i>;<br/><a NAME="E10:27_6_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E11:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E114">A613</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E115">A614</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E13:27_6_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E116">A615</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6_1_2_1_2"><i><font color="Green" title="E108">A607</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E117">A616</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E11:27_6_1_2_1_2"><i><font color="Green" title="E114">A613</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><b>set </b><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E16:27_6_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><a NAME="E17:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E118">A617</font></i>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 ;<br/><a NAME="E18:27_6_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E119">A618</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><i><font color="Green" title="E120">A619</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E12:27_6_1_2_1_2"><i><font color="Green" title="E115">A614</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2

;<br/><a NAME="E21:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E121">A620</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="txt" href="jordan7.html#E19:27_6_1_2_1_2"><i><font color="Green" title="E119">A618</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/><i><font color="Green" title="E122">A621</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E6:27_6_1_2_1_2"><i><font color="Green" title="E111">A610</font></i></a>, <a class="txt" href="jordan7.html#E9:27_6_1_2_1_2"><i><font color="Green" title="E113">A612</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>
;<br/><a NAME="E23:27_6_1_2_1_2"/><i><font color="Green" title="E123">A622</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_6_1_2_1_2"><i><font color="Green" title="E117">A616</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E124">A623</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_6_1_2_1_2"/><i><font color="Green" title="E125">A624</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E1:27_6_1_2_1_2"><i><font color="Green" title="E106">A605</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6_1_2_1_2"><i><font color="Green" title="E107">A606</font></i></a>, <a class="txt" href="jordan7.html#E17:27_6_1_2_1_2"><i><font color="Green" title="E118">A617</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E26:27_6_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E12:27_6_1_2_1_2"><i><font color="Green" title="E115">A614</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E27:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E126">A625</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_6_1_2_1_2"><i><font color="Green" title="E116">A615</font></i></a>, <a class="txt" href="jordan7.html#E20:27_6_1_2_1_2"><i><font color="Green" title="E120">A619</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E127">A626</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_6_1_2_1_2"><i><font color="Green" title="E112">A611</font></i></a>, <a class="txt" href="jordan7.html#E22:27_6_1_2_1_2"><i><font color="Green" title="E122">A621</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E29:27_6_1_2_1_2"/><i><font color="Green" title="E128">A627</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27_6_1_2_1_2"><i><font color="Green" title="E126">A625</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:27_6_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_6"><i><font color="Green" title="E98">A542</font></i></a>, <a class="txt" href="jordan7.html#E4:27_6"><i><font color="Green" title="E101">A545</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E31:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E129">A628</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E32:27_6_1_2_1_2"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_6"><i><font color="Green" title="E99">A543</font></i></a>, <a class="txt" href="jordan7.html#E3:27_6"><i><font color="Green" title="E100">A544</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E33:27_6_1_2_1_2"/><b>then </b><i><font color="Green" title="E130">A629</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E34:27_6_1_2_1_2"/><i><font color="Green" title="E131">A630</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E28:27_6_1_2_1_2"><i><font color="Green" title="E127">A626</font></i></a>, <a class="txt" href="jordan7.html#E29:27_6_1_2_1_2"><i><font color="Green" title="E128">A627</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E35:27_6_1_2_1_2"/><i><font color="Green" title="E132">A631</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E2:27_6_1_2_1_2"><i><font color="Green" title="E107">A606</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self" title="JORDAN7:th.1">Th1</a></i>;<br/><a NAME="E36:27_6_1_2_1_2"/><i><font color="Green" title="E133">A632</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_6_1_2_1_2"><i><font color="Green" title="E112">A611</font></i></a>, <a class="txt" href="jordan7.html#E21:27_6_1_2_1_2"><i><font color="Green" title="E121">A620</font></i></a>, <a class="txt" href="jordan7.html#E22:27_6_1_2_1_2"><i><font color="Green" title="E122">A621</font></i></a>, <a class="txt" href="jordan7.html#E27:27_6_1_2_1_2"><i><font color="Green" title="E126">A625</font></i></a>, <a class="txt" href="jordan7.html#E33:27_6_1_2_1_2"><i><font color="Green" title="E130">A629</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E37:27_6_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/><b>hence </b><a NAME="E38:27_6_1_2_1_2"/>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_6_1_2_1_2"><i><font color="Green" title="E112">A611</font></i></a>, <a class="txt" href="jordan7.html#E22:27_6_1_2_1_2"><i><font color="Green" title="E122">A621</font></i></a>, <a class="txt" href="jordan7.html#E23:27_6_1_2_1_2"><i><font color="Green" title="E123">A622</font></i></a>, <a class="txt" href="jordan7.html#E24:27_6_1_2_1_2"><i><font color="Green" title="E124">A623</font></i></a>, <a class="txt" href="jordan7.html#E25:27_6_1_2_1_2"><i><font color="Green" title="E125">A624</font></i></a>, <a class="txt" href="jordan7.html#E27:27_6_1_2_1_2"><i><font color="Green" title="E126">A625</font></i></a>, <a class="txt" href="jordan7.html#E31:27_6_1_2_1_2"><i><font color="Green" title="E129">A628</font></i></a>, <a class="txt" href="jordan7.html#E33:27_6_1_2_1_2"><i><font color="Green" title="E130">A629</font></i></a>, <a class="txt" href="jordan7.html#E34:27_6_1_2_1_2"><i><font color="Green" title="E131">A630</font></i></a>, <a class="txt" href="jordan7.html#E35:27_6_1_2_1_2"><i><font color="Green" title="E132">A631</font></i></a>, <a class="txt" href="jordan7.html#E36:27_6_1_2_1_2"><i><font color="Green" title="E133">A632</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E100:27"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:27_7"/><i><font color="Green" title="E98">A633</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:27_7"/><b>then </b><i><font color="Green" title="E99">A634</font></i>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a></i>;<br/>
<a NAME="E3:27_7"/><i><font color="Green" title="E100">A635</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E98:27"><i><font color="Green" title="E96">A455</font></i></a>, <a class="txt" href="jordan7.html#E1:27_7"><i><font color="Green" title="E98">A633</font></i></a></i>;<br/>
<a NAME="E4:27_7"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E2:27_7"><i><font color="Green" title="E99">A634</font></i></a>, <a class="ref" href="jordan6.html#T72" title="JORDAN6:th.72">JORDAN6:72</a></i>;<br/>
<b>hence </b><a NAME="E5:27_7"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E2:27_7"><i><font color="Green" title="E99">A634</font></i></a>, <a class="txt" href="jordan7.html#E3:27_7"><i><font color="Green" title="E100">A635</font></i></a>, <a class="ref" href="jordan7.html#T10" target="_self" title="JORDAN7:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E101:27"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8"><b>proof </b></a><div class="add">

<a NAME="E1:27_8"/><i><font color="Green" title="E98">A636</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green" title="E48">A25</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E2:27_8"/><i><font color="Green" title="E99">A637</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green" title="E85">A62</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E3:27_8"/><i><font color="Green" title="E100">A638</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:27_8"/><i><font color="Green" title="E101">A639</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green" title="E84">A61</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E5:27_8"/><b>then </b><i><font color="Green" title="E102">A640</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E2:27_8"><i><font color="Green" title="E99">A637</font></i></a>, <a class="txt" href="jordan7.html#E3:27_8"><i><font color="Green" title="E100">A638</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:27_8"/><i><font color="Green" title="E103">A641</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E63:27"><i><font color="Green" title="E71">A48</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_8"/><i><font color="Green" title="E104">A642</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E8:27_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:27_8"/><b>then </b><i><font color="Green" title="E105">A643</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="jordan7.html#E6:27_8"><i><font color="Green" title="E103">A641</font></i></a>, <a class="txt" href="jordan7.html#E7:27_8"><i><font color="Green" title="E104">A642</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E10:27_8"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E11:27_8"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:27_8"/><b>then </b><i><font color="Green" title="E106">A644</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E13:27_8"/><i><font color="Green" title="E107">A645</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E12:27_8"><i><font color="Green" title="E106">A644</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:27_8"/><b>then </b><i><font color="Green" title="E108">A646</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27_8"><i><font color="Green" title="E105">A643</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<div><i><font color="Green" title="E109">A647</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_8_1"/><i><font color="Green" title="E110">A648</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_8_1"/>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_8"><i><font color="Green" title="E107">A645</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:27_8_1"/><b>then </b>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:27_8_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E5:27_8"><i><font color="Green" title="E102">A640</font></i></a>, <a class="txt" href="jordan7.html#E14:27_8"><i><font color="Green" title="E108">A646</font></i></a>, <a class="txt" href="jordan7.html#E1:27_8_1"><i><font color="Green" title="E110">A648</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:27_8_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E6:27_8_1"/><b>then </b>
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E77:27"><i><font color="Green" title="E85">A62</font></i></a>, <a class="txt" href="jordan7.html#E1:27_8"><i><font color="Green" title="E98">A636</font></i></a>, <a class="txt" href="jordan7.html#E4:27_8"><i><font color="Green" title="E101">A639</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:27_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green" title="E60">A37</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> );<br/>
<a NAME="E16:27_8"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#T65" title="JORDAN6:th.65">JORDAN6:65</a></i>;<br/>
<a NAME="E17:27_8"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_8"><i><font color="Green" title="E102">A640</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E18:27_8"/><b>then </b><i><font color="Green" title="E110">A649</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#T71" title="JORDAN6:th.71">JORDAN6:71</a></i>;<br/>
<a NAME="E19:27_8"/><i><font color="Green" title="E111">A650</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:27_8_2"/><i><font color="Green" title="E112">A651</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_8_2_1"/><i><font color="Green" title="E113">A652</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><a NAME="E2:27_8_2_1"/><i><font color="Green" title="E114">A653</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:27_8_2_1"/><b>then </b><i><font color="Green" title="E115">A654</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_8_2_1"><i><font color="Green" title="E113">A652</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_8_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/><a NAME="E5:27_8_2_1"/><b>then </b>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E98:27"><i><font color="Green" title="E96">A455</font></i></a>, <a class="txt" href="jordan7.html#E2:27_8_2_1"><i><font color="Green" title="E114">A653</font></i></a>, <a class="txt" href="jordan7.html#E3:27_8_2_1"><i><font color="Green" title="E115">A654</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E6:27_8_2_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E1:27_8_2"><i><font color="Green" title="E112">A651</font></i></a>, <a class="txt" href="jordan7.html#E1:27_8_2_1"><i><font color="Green" title="E113">A652</font></i></a>, <a class="ref" href="jordan6.html#T73" title="JORDAN6:th.73">JORDAN6:73</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_8_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:27_8"/><i><font color="Green" title="E112">A655</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan7.html#E18:27_8"><i><font color="Green" title="E110">A649</font></i></a>, <a class="txt" href="jordan7.html#E19:27_8"><i><font color="Green" title="E111">A650</font></i></a>);<br/></i>
<a NAME="E21:27_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:27_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
;<br/>
<a NAME="E23:27_8"/><b>then </b>
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green" title="E84">A61</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green" title="E85">A62</font></i></a>, <a class="txt" href="jordan7.html#E1:27_8"><i><font color="Green" title="E98">A636</font></i></a>, <a class="txt" href="jordan7.html#E15:27_8"><i><font color="Green" title="E109">A647</font></i></a>, <a class="txt" href="jordan7.html#E20:27_8"><i><font color="Green" title="E112">A655</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E24:27_8"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green" title="E88">A65</font></i></a>, <a class="ref" href="jordan7.html#T12" target="_self" title="JORDAN7:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon" title="c10">i</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E102:27"/><i><font color="Green" title="E98">A656</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green" title="E47">A24</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E103:27"/><i><font color="Green" title="E99">A657</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="jordan7.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E104:27"/><i><font color="Green" title="E100">A658</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E86:27"><i><font color="Green" title="E93">A126</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E105:27"/><b>then </b><i><font color="Green" title="E101">A659</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E106:27"/><i><font color="Green" title="E102">A660</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green" title="E76">A53</font></i></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></i>;<br/>
<a NAME="E107:27"/><b>then </b><i><font color="Green" title="E103">A661</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E103:27"><i><font color="Green" title="E99">A657</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E108:27"/><b>then </b><i><font color="Green" title="E104">A662</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E109:27"/><i><font color="Green" title="E105">A663</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E110:27"/><i><font color="Green" title="E106">A664</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green" title="E76">A53</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E111:27"/><i><font color="Green" title="E107">A665</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E103:27"><i><font color="Green" title="E99">A657</font></i></a>, <a class="txt" href="jordan7.html#E109:27"><i><font color="Green" title="E105">A663</font></i></a></i>;<br/>
<a NAME="E112:27"/><i><font color="Green" title="E108">A666</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E110:27"><i><font color="Green" title="E106">A664</font></i></a></i>;<br/>
<a NAME="E113:27"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E110:27"><i><font color="Green" title="E106">A664</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E114:27"/><b>then </b><i><font color="Green" title="E109">A667</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E115:27"/><i><font color="Green" title="E110">A668</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E110:27"><i><font color="Green" title="E106">A664</font></i></a></i>;<br/>
<a NAME="E116:27"/><i><font color="Green" title="E111">A669</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E117:27"/><i><font color="Green" title="E112">A670</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E118:27"/><b>then </b><i><font color="Green" title="E113">A671</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E116:27"><i><font color="Green" title="E111">A669</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E119:27"/><i><font color="Green" title="E114">A672</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E117:27"><i><font color="Green" title="E112">A670</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E120:27"/><i><font color="Green" title="E115">A673</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E105:27"><i><font color="Green" title="E101">A659</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green" title="E116">A674</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_9_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1_1"><b>case </b></a><a NAME="E1:27_9_1_1"/><i><font color="Green" title="E117">A675</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_9_1_1"/><b>then </b><i><font color="Green" title="E118">A676</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E120:27"><i><font color="Green" title="E115">A673</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_9_1_1"/><i><font color="Green" title="E119">A677</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E120:27"><i><font color="Green" title="E115">A673</font></i></a>, <a class="txt" href="jordan7.html#E1:27_9_1_1"><i><font color="Green" title="E117">A675</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_9_1_1"/><b>then </b><i><font color="Green" title="E120">A678</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E105:27"><i><font color="Green" title="E101">A659</font></i></a>, <a class="txt" href="jordan7.html#E117:27"><i><font color="Green" title="E112">A670</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:27_9_1_1"/><i><font color="Green" title="E121">A679</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_9_1_1"><i><font color="Green" title="E119">A677</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_9_1_1"/><i><font color="Green" title="E122">A680</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_9_1_1"><i><font color="Green" title="E118">A676</font></i></a>, <a class="txt" href="jordan7.html#E3:27_9_1_1"><i><font color="Green" title="E119">A677</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E7:27_9_1_1"/><i><font color="Green" title="E123">A681</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E5:27_9_1_1"><i><font color="Green" title="E121">A679</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_9_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E119:27"><i><font color="Green" title="E114">A672</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_9_1_1"/><b>then </b>
( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E108:27"><i><font color="Green" title="E104">A662</font></i></a>, <a class="txt" href="jordan7.html#E114:27"><i><font color="Green" title="E109">A667</font></i></a>, <a class="txt" href="jordan7.html#E118:27"><i><font color="Green" title="E113">A671</font></i></a>, <a class="txt" href="jordan7.html#E119:27"><i><font color="Green" title="E114">A672</font></i></a>, <a class="txt" href="jordan7.html#E4:27_9_1_1"><i><font color="Green" title="E120">A678</font></i></a>, <a class="txt" href="jordan7.html#E5:27_9_1_1"><i><font color="Green" title="E121">A679</font></i></a>, <a class="txt" href="jordan7.html#E6:27_9_1_1"><i><font color="Green" title="E122">A680</font></i></a>, <a class="txt" href="jordan7.html#E7:27_9_1_1"><i><font color="Green" title="E123">A681</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:27_9_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1_2"><b>case </b></a><a NAME="E1:27_9_1_2"/><i><font color="Green" title="E117">A682</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_9_1_2"/><b>then </b><i><font color="Green" title="E118">A683</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E3:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:27_9_1_2"/><b>then </b><i><font color="Green" title="E119">A684</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_9_1_2"><i><font color="Green" title="E117">A682</font></i></a>, <a class="txt" href="jordan7.html#E2:27_9_1_2"><i><font color="Green" title="E118">A683</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E5:27_9_1_2"/><i><font color="Green" title="E120">A685</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_9_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_9_1_2"><i><font color="Green" title="E119">A684</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:27_9_1_2"/><b>then </b><i><font color="Green" title="E121">A686</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E8:27_9_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E9:27_9_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_9_1_2"/><b>then </b><i><font color="Green" title="E122">A687</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E11:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_9_1_2"><i><font color="Green" title="E118">A683</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E14:27_9_1_2"/><b>then </b><i><font color="Green" title="E123">A688</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_9_1_2"><i><font color="Green" title="E122">A687</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:27_9_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_9_1_2"/><b>then </b><i><font color="Green" title="E124">A689</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E17:27_9_1_2"/><i><font color="Green" title="E125">A690</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_9_1_2"><i><font color="Green" title="E117">A682</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:27_9_1_2"/><b>then </b><i><font color="Green" title="E126">A691</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E19:27_9_1_2"/><i><font color="Green" title="E127">A692</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_9_1_2"><i><font color="Green" title="E125">A690</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_9_1_2"/><b>then </b><i><font color="Green" title="E128">A693</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E21:27_9_1_2"/><i><font color="Green" title="E129">A694</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E2:27_9_1_2"><i><font color="Green" title="E118">A683</font></i></a>, <a class="txt" href="jordan7.html#E5:27_9_1_2"><i><font color="Green" title="E120">A685</font></i></a>, <a class="txt" href="jordan7.html#E18:27_9_1_2"><i><font color="Green" title="E126">A691</font></i></a>, <a class="txt" href="jordan7.html#E19:27_9_1_2"><i><font color="Green" title="E127">A692</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E22:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E109:27"><i><font color="Green" title="E105">A663</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E24:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_9_1_2"><i><font color="Green" title="E118">A683</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E25:27_9_1_2"/><b>then </b><i><font color="Green" title="E130">A695</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon" title="c11">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E26:27_9_1_2"/><i><font color="Green" title="E131">A696</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E14:27_9_1_2"><i><font color="Green" title="E123">A688</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:27_9_1_2"/><i><font color="Green" title="E132">A697</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green" title="E82">A59</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E28:27_9_1_2"/><i><font color="Green" title="E133">A698</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E2:27_9_1_2"><i><font color="Green" title="E118">A683</font></i></a>, <a class="txt" href="jordan7.html#E5:27_9_1_2"><i><font color="Green" title="E120">A685</font></i></a>, <a class="txt" href="jordan7.html#E18:27_9_1_2"><i><font color="Green" title="E126">A691</font></i></a>, <a class="txt" href="jordan7.html#E20:27_9_1_2"><i><font color="Green" title="E128">A693</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E29:27_9_1_2"/><i><font color="Green" title="E134">A699</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E14:27_9_1_2"><i><font color="Green" title="E123">A688</font></i></a>, <a class="txt" href="jordan7.html#E21:27_9_1_2"><i><font color="Green" title="E129">A694</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E30:27_9_1_2"/><i><font color="Green" title="E135">A700</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E27:27_9_1_2"><i><font color="Green" title="E132">A697</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_9_1_2"/><i><font color="Green" title="E136">A701</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E26:27_9_1_2"><i><font color="Green" title="E131">A696</font></i></a>, <a class="txt" href="jordan7.html#E29:27_9_1_2"><i><font color="Green" title="E134">A699</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_9_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E108:27"><i><font color="Green" title="E104">A662</font></i></a>, <a class="txt" href="jordan7.html#E7:27_9_1_2"><i><font color="Green" title="E121">A686</font></i></a>, <a class="txt" href="jordan7.html#E14:27_9_1_2"><i><font color="Green" title="E123">A688</font></i></a>, <a class="txt" href="jordan7.html#E25:27_9_1_2"><i><font color="Green" title="E130">A695</font></i></a>, <a class="txt" href="jordan7.html#E28:27_9_1_2"><i><font color="Green" title="E133">A698</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E33:27_9_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E108:27"><i><font color="Green" title="E104">A662</font></i></a>, <a class="txt" href="jordan7.html#E114:27"><i><font color="Green" title="E109">A667</font></i></a>, <a class="txt" href="jordan7.html#E116:27"><i><font color="Green" title="E111">A669</font></i></a>, <a class="txt" href="jordan7.html#E16:27_9_1_2"><i><font color="Green" title="E124">A689</font></i></a>, <a class="txt" href="jordan7.html#E30:27_9_1_2"><i><font color="Green" title="E135">A700</font></i></a>, <a class="txt" href="jordan7.html#E31:27_9_1_2"><i><font color="Green" title="E136">A701</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E122:27"/><i><font color="Green" title="E117">A702</font></i>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E102:27"><i><font color="Green" title="E98">A656</font></i></a>, <a class="txt" href="jordan7.html#E103:27"><i><font color="Green" title="E99">A657</font></i></a></i>;<br/>
<a NAME="E123:27"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E104:27"><i><font color="Green" title="E100">A658</font></i></a>, <a class="txt" href="jordan7.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E124:27"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green" title="E91">A68</font></i></a>, <a class="txt" href="jordan7.html#E107:27"><i><font color="Green" title="E103">A661</font></i></a>, <a class="ref" href="partfun2.html#T17" title="PARTFUN2:th.17">PARTFUN2:17</a></i>;<br/>
<b>hence </b><a NAME="E125:27"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green" title="E88">A65</font></i></a>, <a class="txt" href="jordan7.html#E102:27"><i><font color="Green" title="E98">A656</font></i></a>, <a class="txt" href="jordan7.html#E122:27"><i><font color="Green" title="E117">A702</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self" title="JORDAN7:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font> &amp; ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_10"/><i><font color="Green" title="E118">A703</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_10"/><i><font color="Green" title="E119">A704</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_10"/><b>then </b><i><font color="Green" title="E120">A705</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:27_10"/><i><font color="Green" title="E121">A706</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:27_10"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_10"><i><font color="Green" title="E119">A704</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_10"/><b>then </b><i><font color="Green" title="E122">A707</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E3:27_10"><i><font color="Green" title="E120">A705</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_10_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1_1"><b>suppose </b></a><a NAME="E1:27_10_1_1"/><i><font color="Green" title="E123">A708</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_10_1_1"/><b>then </b><i><font color="Green" title="E124">A709</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="jordan7.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_10_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E103:27"><i><font color="Green" title="E99">A657</font></i></a>, <a class="txt" href="jordan7.html#E1:27_10_1_1"><i><font color="Green" title="E123">A708</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:27_10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E105:27"><i><font color="Green" title="E101">A659</font></i></a>, <a class="txt" href="jordan7.html#E108:27"><i><font color="Green" title="E104">A662</font></i></a>, <a class="txt" href="jordan7.html#E114:27"><i><font color="Green" title="E109">A667</font></i></a>, <a class="txt" href="jordan7.html#E1:27_10"><i><font color="Green" title="E118">A703</font></i></a>, <a class="txt" href="jordan7.html#E2:27_10"><i><font color="Green" title="E119">A704</font></i></a>, <a class="txt" href="jordan7.html#E4:27_10"><i><font color="Green" title="E121">A706</font></i></a>, <a class="txt" href="jordan7.html#E2:27_10_1_1"><i><font color="Green" title="E124">A709</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1_2"><b>suppose </b></a><a NAME="E1:27_10_1_2"/><i><font color="Green" title="E123">A710</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_10_1_2"/><b>then </b><i><font color="Green" title="E124">A711</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_10_1_2"/><b>then </b><i><font color="Green" title="E125">A712</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_10_1_2"/><i><font color="Green" title="E126">A713</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_10_1_2"><i><font color="Green" title="E123">A710</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:27_10_1_2"/><i><font color="Green" title="E127">A714</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_10_1_2"><i><font color="Green" title="E124">A711</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_10_1_2"/><b>then </b><i><font color="Green" title="E128">A715</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_10_1_2"><i><font color="Green" title="E125">A712</font></i></a>, <a class="txt" href="jordan7.html#E4:27_10_1_2"><i><font color="Green" title="E126">A713</font></i></a>, <a class="txt" href="jordan7.html#E5:27_10_1_2"><i><font color="Green" title="E127">A714</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_10_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E9:27_10_1_2"/><b>then </b><i><font color="Green" title="E129">A716</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_10_1_2"/><b>then </b><i><font color="Green" title="E130">A717</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E11:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_10_1_2"><i><font color="Green" title="E126">A713</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_10_1_2"/><b>then </b><i><font color="Green" title="E131">A718</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E9:27_10_1_2"><i><font color="Green" title="E129">A716</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E14:27_10_1_2"/><i><font color="Green" title="E132">A719</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_10_1_2"><i><font color="Green" title="E130">A717</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c11">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E15:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_10_1_2"/><b>then </b><i><font color="Green" title="E133">A720</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_10_1_2"><i><font color="Green" title="E126">A713</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E17:27_10_1_2"/><i><font color="Green" title="E134">A721</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_10_1_2"><i><font color="Green" title="E132">A719</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_10_1_2"><i><font color="Green" title="E124">A711</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:27_10_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E20:27_10_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_10_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_10_1_2"/><b>then </b><i><font color="Green" title="E135">A722</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_10_1_2"><i><font color="Green" title="E132">A719</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E23:27_10_1_2"/><i><font color="Green" title="E136">A723</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E13:27_10_1_2"><i><font color="Green" title="E131">A718</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_10_1_2"/><b>then </b><i><font color="Green" title="E137">A724</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27_10_1_2"><i><font color="Green" title="E128">A715</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_10_1_2"/><i><font color="Green" title="E138">A725</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E23:27_10_1_2"><i><font color="Green" title="E136">A723</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E26:27_10_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E103:27"><i><font color="Green" title="E99">A657</font></i></a>, <a class="txt" href="jordan7.html#E106:27"><i><font color="Green" title="E102">A660</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_10_1_2"/><b>then </b><i><font color="Green" title="E139">A726</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E28:27_10_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E25:27_10_1_2"><i><font color="Green" title="E138">A725</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_10_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E114:27"><i><font color="Green" title="E109">A667</font></i></a>, <a class="txt" href="jordan7.html#E1:27_10"><i><font color="Green" title="E118">A703</font></i></a>, <a class="txt" href="jordan7.html#E6:27_10"><i><font color="Green" title="E122">A707</font></i></a>, <a class="txt" href="jordan7.html#E24:27_10_1_2"><i><font color="Green" title="E137">A724</font></i></a>, <a class="txt" href="jordan7.html#E27:27_10_1_2"><i><font color="Green" title="E139">A726</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E30:27_10_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E31:27_10_1_2"/><b>then </b>
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E32:27_10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E7:27_10_1_2"><i><font color="Green" title="E128">A715</font></i></a>, <a class="txt" href="jordan7.html#E16:27_10_1_2"><i><font color="Green" title="E133">A720</font></i></a>, <a class="txt" href="jordan7.html#E17:27_10_1_2"><i><font color="Green" title="E134">A721</font></i></a>, <a class="txt" href="jordan7.html#E22:27_10_1_2"><i><font color="Green" title="E135">A722</font></i></a>, <a class="txt" href="jordan7.html#E23:27_10_1_2"><i><font color="Green" title="E136">A723</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E127:27"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E102:27"><i><font color="Green" title="E98">A656</font></i></a>, <a class="txt" href="jordan7.html#E111:27"><i><font color="Green" title="E107">A665</font></i></a>, <a class="txt" href="jordan7.html#E112:27"><i><font color="Green" title="E108">A666</font></i></a>, <a class="txt" href="jordan7.html#E115:27"><i><font color="Green" title="E110">A668</font></i></a>, <a class="txt" href="jordan7.html#E121:27"><i><font color="Green" title="E116">A674</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self" title="JORDAN7:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> ) )</span><br/>

<b>thus </b><a NAME="E128:27"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> holds <br/> <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> &amp; not <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a>  implies  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_11"/><i><font color="Green" title="E118">A727</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E2:27_11"/><i><font color="Green" title="E119">A728</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 <b>and </b><br/><a NAME="E3:27_11"/><i><font color="Green" title="E120">A729</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 <b>and </b><br/><a NAME="E4:27_11"/><i><font color="Green" title="E121">A730</font></i>: 
not <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font> <a href="gobrd10.html#R1" title="GOBRD10:pred.1">are_adjacent1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E5:27_11"/><i><font color="Green" title="E122">A731</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27_11"><i><font color="Green" title="E119">A728</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:27_11"/><i><font color="Green" title="E123">A732</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11"><i><font color="Green" title="E119">A728</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_11"/><i><font color="Green" title="E124">A733</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27_11"><i><font color="Green" title="E122">A731</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E8:27_11"/><i><font color="Green" title="E125">A734</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E9:27_11"/><i><font color="Green" title="E126">A735</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a></i>;<br/>
<a NAME="E10:27_11"/><i><font color="Green" title="E127">A736</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11"><i><font color="Green" title="E125">A734</font></i></a></i>;<br/>
<a NAME="E11:27_11"/><i><font color="Green" title="E128">A737</font></i>: 
( not <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; not <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E4:27_11"><i><font color="Green" title="E121">A730</font></i></a>, <a class="ref" href="gobrd10.html#D1" title="GOBRD10:def.1">GOBRD10:def 1</a></i>;<br/>
<a NAME="E12:27_11"/><i><font color="Green" title="E129">A738</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27_11"><i><font color="Green" title="E119">A728</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E13:27_11"/><b>then </b><i><font color="Green" title="E130">A739</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27_11"><i><font color="Green" title="E128">A737</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E14:27_11"/><i><font color="Green" title="E131">A740</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:27_11"/><i><font color="Green" title="E132">A741</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a></i>;<br/>
<a NAME="E16:27_11"/><i><font color="Green" title="E133">A742</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E17:27_11"/><i><font color="Green" title="E134">A743</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:27_11"/><i><font color="Green" title="E135">A744</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_1_1_1"/><i><font color="Green" title="E136">A745</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_1_1_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_1_1_1_1_1"/><i><font color="Green" title="E137">A746</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_11_1_1_1_1_1"/><i><font color="Green" title="E138">A747</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1"><i><font color="Green" title="E136">A745</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_11_1_1_1_1_1"/><i><font color="Green" title="E139">A748</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1"><i><font color="Green" title="E136">A745</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E140">A749</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:27_11_1_1_1_1_1"/><i><font color="Green" title="E141">A750</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_11"><i><font color="Green" title="E130">A739</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E142">A751</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_1"><i><font color="Green" title="E137">A746</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_11_1_1_1_1_1"/><i><font color="Green" title="E143">A752</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_1"><i><font color="Green" title="E137">A746</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_1_1_1_1_1"><i><font color="Green" title="E141">A750</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E144">A753</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_11_1_1_1_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E145">A754</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E11:27_11_1_1_1_1_1"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_1_1_1_1_1"><i><font color="Green" title="E141">A750</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:27_11_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E13:27_11_1_1_1_1_1"/><b>then </b>
(  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">j</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E13:27_11"><i><font color="Green" title="E130">A739</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_1_1_1_1_1"><i><font color="Green" title="E138">A747</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_1_1_1"><i><font color="Green" title="E139">A748</font></i></a>, <a class="txt" href="jordan7.html#E4:27_11_1_1_1_1_1"><i><font color="Green" title="E140">A749</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_1_1_1_1_1"><i><font color="Green" title="E142">A751</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11_1_1_1_1_1"><i><font color="Green" title="E143">A752</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_1_1_1_1_1"><i><font color="Green" title="E144">A753</font></i></a>, <a class="txt" href="jordan7.html#E10:27_11_1_1_1_1_1"><i><font color="Green" title="E145">A754</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a>, <a class="ref" href="jordan6.html#D8" title="JORDAN6:def.8">JORDAN6:def 8</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_11_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>, <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E4:27_11_1_1_1_1_1"><i><font color="Green" title="E140">A749</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_1_1_1_1_1"><i><font color="Green" title="E144">A753</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E15:27_11_1_1_1_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_1_1_1_1_2"/><i><font color="Green" title="E137">A755</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E138">A756</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:27_11_1_1_1_1_2"/><i><font color="Green" title="E139">A757</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_2"><i><font color="Green" title="E137">A755</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E4:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_11_1_1_1_1_2"><i><font color="Green" title="E138">A756</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E140">A758</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_2"><i><font color="Green" title="E137">A755</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_1_1_2"><i><font color="Green" title="E139">A757</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E6:27_11_1_1_1_1_2"/><i><font color="Green" title="E141">A759</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_1_1_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_11_1_1_1_1_2"><i><font color="Green" title="E140">A758</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E142">A760</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E9:27_11_1_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E10:27_11_1_1_1_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E143">A761</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E12:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_1_1_2"><i><font color="Green" title="E139">A757</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E144">A762</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E15:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E145">A763</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_11_1_1_1_1_2"><i><font color="Green" title="E143">A761</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1"><i><font color="Green" title="E136">A745</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:27_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E146">A764</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font> &amp; <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E16:27_11"><i><font color="Green" title="E133">A742</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1"><i><font color="Green" title="E136">A745</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_2"><i><font color="Green" title="E137">A755</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_1_1_1_1_2"><i><font color="Green" title="E141">A759</font></i></a>, <a class="txt" href="jordan7.html#E11:27_11_1_1_1_1_2"><i><font color="Green" title="E143">A761</font></i></a>, <a class="txt" href="jordan7.html#E14:27_11_1_1_1_1_2"><i><font color="Green" title="E144">A762</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E19:27_11_1_1_1_1_2"/><i><font color="Green" title="E147">A765</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11_1_1_1_1_2"><i><font color="Green" title="E137">A755</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E20:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E148">A766</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E21:27_11_1_1_1_1_2"/><i><font color="Green" title="E149">A767</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E19:27_11_1_1_1_1_2"><i><font color="Green" title="E147">A765</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E150">A768</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_1_1_2"><i><font color="Green" title="E139">A757</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_1_1_1_1_2"><i><font color="Green" title="E141">A759</font></i></a>, <a class="txt" href="jordan7.html#E20:27_11_1_1_1_1_2"><i><font color="Green" title="E148">A766</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E23:27_11_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11"><i><font color="Green" title="E125">A734</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E24:27_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E25:27_11_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_1_1_2"><i><font color="Green" title="E139">A757</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E26:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E151">A769</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_1_1_1_1_2"/><i><font color="Green" title="E152">A770</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22"><i><font color="Green" title="E19">Lm4</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E28:27_11_1_1_1_1_2"/><i><font color="Green" title="E153">A771</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_1_1_2"><i><font color="Green" title="E145">A763</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27_11"><i><font color="Green" title="E133">A742</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_1"><i><font color="Green" title="E136">A745</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E154">A772</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_11_1_1_1_1_2"/><i><font color="Green" title="E155">A773</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E18:27_11_1_1_1_1_2"><i><font color="Green" title="E146">A764</font></i></a>, <a class="txt" href="jordan7.html#E20:27_11_1_1_1_1_2"><i><font color="Green" title="E148">A766</font></i></a>, <a class="txt" href="jordan7.html#E21:27_11_1_1_1_1_2"><i><font color="Green" title="E149">A767</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E32:27_11_1_1_1_1_2"/><i><font color="Green" title="E156">A774</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_1_1_2"><i><font color="Green" title="E145">A763</font></i></a>, <a class="txt" href="jordan7.html#E22:27_11_1_1_1_1_2"><i><font color="Green" title="E150">A768</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E33:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E157">A775</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E35:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E36:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E158">A776</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E37:27_11_1_1_1_1_2"/><i><font color="Green" title="E159">A777</font></i>: 
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E30:27_11_1_1_1_1_2"><i><font color="Green" title="E154">A772</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E38:27_11_1_1_1_1_2"/><i><font color="Green" title="E160">A778</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E28:27_11_1_1_1_1_2"><i><font color="Green" title="E153">A771</font></i></a>, <a class="txt" href="jordan7.html#E32:27_11_1_1_1_1_2"><i><font color="Green" title="E156">A774</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E39:27_11_1_1_1_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_1_1_1_1_2"><i><font color="Green" title="E142">A760</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_1_1_2"><i><font color="Green" title="E145">A763</font></i></a>, <a class="txt" href="jordan7.html#E26:27_11_1_1_1_1_2"><i><font color="Green" title="E151">A769</font></i></a>, <a class="txt" href="jordan7.html#E27:27_11_1_1_1_1_2"><i><font color="Green" title="E152">A770</font></i></a>, <a class="txt" href="jordan7.html#E31:27_11_1_1_1_1_2"><i><font color="Green" title="E155">A773</font></i></a>, <a class="txt" href="jordan7.html#E36:27_11_1_1_1_1_2"><i><font color="Green" title="E158">A776</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E40:27_11_1_1_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E18:27_11_1_1_1_1_2"><i><font color="Green" title="E146">A764</font></i></a>, <a class="txt" href="jordan7.html#E34:27_11_1_1_1_1_2"><i><font color="Green" title="E157">A775</font></i></a>, <a class="txt" href="jordan7.html#E36:27_11_1_1_1_1_2"><i><font color="Green" title="E158">A776</font></i></a>, <a class="txt" href="jordan7.html#E37:27_11_1_1_1_1_2"><i><font color="Green" title="E159">A777</font></i></a>, <a class="txt" href="jordan7.html#E38:27_11_1_1_1_1_2"><i><font color="Green" title="E160">A778</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_1_1_2"/><i><font color="Green" title="E136">A779</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E137">A780</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:27_11_1_1_2"/><i><font color="Green" title="E138">A781</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_2"><i><font color="Green" title="E136">A779</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E4:27_11_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E139">A782</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_1_1_2"><i><font color="Green" title="E137">A780</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_2"><i><font color="Green" title="E138">A781</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E6:27_11_1_1_2"/><i><font color="Green" title="E140">A783</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_11_1_1_2"><i><font color="Green" title="E139">A782</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E141">A784</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E9:27_11_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E10:27_11_1_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E142">A785</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E12:27_11_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_2"><i><font color="Green" title="E138">A781</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E143">A786</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E15:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E144">A787</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_11_1_1_2"><i><font color="Green" title="E142">A785</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_11_1_1_2"/><i><font color="Green" title="E145">A788</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E14:27_11"><i><font color="Green" title="E131">A740</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_2"><i><font color="Green" title="E136">A779</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_11_1_1_2"/><i><font color="Green" title="E146">A789</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E14:27_11"><i><font color="Green" title="E131">A740</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_11_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11_1_1_2"><i><font color="Green" title="E136">A779</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E19:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E147">A790</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><i><font color="Green" title="E148">A791</font></i>: <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E16:27_11_1_1_2"><i><font color="Green" title="E145">A788</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11_1_1_2"><i><font color="Green" title="E146">A789</font></i></a>, <a class="txt" href="jordan7.html#E20:27_11_1_1_2"><i><font color="Green" title="E147">A790</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E22:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E149">A792</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> &amp; <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_2"><i><font color="Green" title="E136">A779</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_1_1_2"><i><font color="Green" title="E137">A780</font></i></a>, <a class="txt" href="jordan7.html#E11:27_11_1_1_2"><i><font color="Green" title="E142">A785</font></i></a>, <a class="txt" href="jordan7.html#E14:27_11_1_1_2"><i><font color="Green" title="E143">A786</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11_1_1_2"><i><font color="Green" title="E146">A789</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_11_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_1_1_2"><i><font color="Green" title="E136">A779</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E24:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E150">A793</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E25:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E151">A794</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E26:27_11_1_1_2"/><i><font color="Green" title="E152">A795</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E24:27_11_1_1_2"><i><font color="Green" title="E150">A793</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E153">A796</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_2"><i><font color="Green" title="E138">A781</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_1_1_2"><i><font color="Green" title="E140">A783</font></i></a>, <a class="txt" href="jordan7.html#E25:27_11_1_1_2"><i><font color="Green" title="E151">A794</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E28:27_11_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E29:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E154">A797</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E30:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E155">A798</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E31:27_11_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E22:27_11_1_1_2"><i><font color="Green" title="E149">A792</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E32:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E156">A799</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E29:27_11_1_1_2"><i><font color="Green" title="E154">A797</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E33:27_11_1_1_2"/><i><font color="Green" title="E157">A800</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E30:27_11_1_1_2"><i><font color="Green" title="E155">A798</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E34:27_11_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11"><i><font color="Green" title="E125">A734</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E35:27_11_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E36:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_1_1_2"><i><font color="Green" title="E138">A781</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E37:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E158">A801</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E38:27_11_1_1_2"/><i><font color="Green" title="E159">A802</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22"><i><font color="Green" title="E19">Lm4</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>set </b><font color="Maroon" title="c14">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E39:27_11_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_11"><i><font color="Green" title="E130">A739</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E40:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27_11_1_1_2"><i><font color="Green" title="E149">A792</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E41:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E160">A803</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_11_1_1_2"><i><font color="Green" title="E141">A784</font></i></a>, <a class="txt" href="jordan7.html#E33:27_11_1_1_2"><i><font color="Green" title="E157">A800</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E42:27_11_1_1_2"/><i><font color="Green" title="E161">A804</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_2"><i><font color="Green" title="E144">A787</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E43:27_11_1_1_2"/><i><font color="Green" title="E162">A805</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_2"><i><font color="Green" title="E144">A787</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E44:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E163">A806</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E45:27_11_1_1_2"/><i><font color="Green" title="E164">A807</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27_11_1_1_2"><i><font color="Green" title="E156">A799</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E46:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E165">A808</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E21:27_11_1_1_2"><i><font color="Green" title="E148">A791</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E47:27_11_1_1_2"/><i><font color="Green" title="E166">A809</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E45:27_11_1_1_2"><i><font color="Green" title="E164">A807</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E48:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E167">A810</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E49:27_11_1_1_2"/><i><font color="Green" title="E168">A811</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_2"><i><font color="Green" title="E144">A787</font></i></a>, <a class="txt" href="jordan7.html#E41:27_11_1_1_2"><i><font color="Green" title="E160">A803</font></i></a>, <a class="txt" href="jordan7.html#E45:27_11_1_1_2"><i><font color="Green" title="E164">A807</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E50:27_11_1_1_2"/><i><font color="Green" title="E169">A812</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E22:27_11_1_1_2"><i><font color="Green" title="E149">A792</font></i></a>, <a class="txt" href="jordan7.html#E25:27_11_1_1_2"><i><font color="Green" title="E151">A794</font></i></a>, <a class="txt" href="jordan7.html#E26:27_11_1_1_2"><i><font color="Green" title="E152">A795</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E51:27_11_1_1_2"/><i><font color="Green" title="E170">A813</font></i>: 
( <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_2"><i><font color="Green" title="E144">A787</font></i></a>, <a class="txt" href="jordan7.html#E27:27_11_1_1_2"><i><font color="Green" title="E153">A796</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E52:27_11_1_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E17:27_11"><i><font color="Green" title="E134">A743</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E53:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E171">A814</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E54:27_11_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E55:27_11_1_1_2"/><b>then </b><i><font color="Green" title="E172">A815</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E56:27_11_1_1_2"/><i><font color="Green" title="E173">A816</font></i>: 
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E47:27_11_1_1_2"><i><font color="Green" title="E166">A809</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E57:27_11_1_1_2"/><i><font color="Green" title="E174">A817</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E43:27_11_1_1_2"><i><font color="Green" title="E162">A805</font></i></a>, <a class="txt" href="jordan7.html#E51:27_11_1_1_2"><i><font color="Green" title="E170">A813</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E58:27_11_1_1_2"/><i><font color="Green" title="E175">A818</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_1_1_2"><i><font color="Green" title="E141">A784</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_1_1_2"><i><font color="Green" title="E144">A787</font></i></a>, <a class="txt" href="jordan7.html#E37:27_11_1_1_2"><i><font color="Green" title="E158">A801</font></i></a>, <a class="txt" href="jordan7.html#E38:27_11_1_1_2"><i><font color="Green" title="E159">A802</font></i></a>, <a class="txt" href="jordan7.html#E50:27_11_1_1_2"><i><font color="Green" title="E169">A812</font></i></a>, <a class="txt" href="jordan7.html#E55:27_11_1_1_2"><i><font color="Green" title="E172">A815</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E59:27_11_1_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E27:27_11_1_1_2"><i><font color="Green" title="E153">A796</font></i></a>, <a class="txt" href="jordan7.html#E42:27_11_1_1_2"><i><font color="Green" title="E161">A804</font></i></a>, <a class="txt" href="jordan7.html#E44:27_11_1_1_2"><i><font color="Green" title="E163">A806</font></i></a>, <a class="txt" href="jordan7.html#E46:27_11_1_1_2"><i><font color="Green" title="E165">A808</font></i></a>, <a class="txt" href="jordan7.html#E48:27_11_1_1_2"><i><font color="Green" title="E167">A810</font></i></a>, <a class="txt" href="jordan7.html#E49:27_11_1_1_2"><i><font color="Green" title="E168">A811</font></i></a>, <a class="txt" href="jordan7.html#E53:27_11_1_1_2"><i><font color="Green" title="E171">A814</font></i></a>, <a class="txt" href="jordan7.html#E55:27_11_1_1_2"><i><font color="Green" title="E172">A815</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E60:27_11_1_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E46:27_11_1_1_2"><i><font color="Green" title="E165">A808</font></i></a>, <a class="txt" href="jordan7.html#E53:27_11_1_1_2"><i><font color="Green" title="E171">A814</font></i></a>, <a class="txt" href="jordan7.html#E55:27_11_1_1_2"><i><font color="Green" title="E172">A815</font></i></a>, <a class="txt" href="jordan7.html#E56:27_11_1_1_2"><i><font color="Green" title="E173">A816</font></i></a>, <a class="txt" href="jordan7.html#E57:27_11_1_1_2"><i><font color="Green" title="E174">A817</font></i></a>, <a class="txt" href="jordan7.html#E58:27_11_1_1_2"><i><font color="Green" title="E175">A818</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_11_2"/><i><font color="Green" title="E136">A819</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_2_1"/>
( <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1"><b>suppose </b></a><a NAME="E1:27_11_2_1_1"/><i><font color="Green" title="E137">A820</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_11_2_1_1"/><i><font color="Green" title="E138">A821</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_11_2_1_1"/><b>then </b><i><font color="Green" title="E139">A822</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11_2_1_1"><i><font color="Green" title="E137">A820</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_2_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_1"><i><font color="Green" title="E138">A821</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_11_2_1_1"/><b>then </b><i><font color="Green" title="E140">A823</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E6:27_11_2_1_1"/><i><font color="Green" title="E141">A824</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_1"><i><font color="Green" title="E137">A820</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_1"><i><font color="Green" title="E138">A821</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_11_2_1_1"/><b>then </b><i><font color="Green" title="E142">A825</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11_2_1_1"><i><font color="Green" title="E139">A822</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E8:27_11_2_1_1"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11_2_1_1"><i><font color="Green" title="E139">A822</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_11_2_1_1"/><b>then </b><i><font color="Green" title="E143">A826</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11_2_1_1"><i><font color="Green" title="E142">A825</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_2_1_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_2_1_1_1_1"/><i><font color="Green" title="E144">A827</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_11_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E145">A828</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_11_2_1_1_1_1"/><i><font color="Green" title="E146">A829</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_1_1_1"><i><font color="Green" title="E144">A827</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_2_1_1_1_1"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_11_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:27_11_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green" title="E43">A20</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green" title="E89">A66</font></i></a>, <a class="txt" href="jordan7.html#E11:27_11"><i><font color="Green" title="E128">A737</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2"><i><font color="Green" title="E136">A819</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_2_1_1"><i><font color="Green" title="E139">A822</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_1"><i><font color="Green" title="E140">A823</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_2_1_1"><i><font color="Green" title="E141">A824</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_1_1_1"><i><font color="Green" title="E145">A828</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_2_1_1_1_1"><i><font color="Green" title="E146">A829</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_2_1_1_1_2"/><i><font color="Green" title="E144">A830</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E145">A831</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_11_2_1_1_1_2"/><i><font color="Green" title="E146">A832</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_1_1_2"><i><font color="Green" title="E144">A830</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:27_11_2_1_1_1_2"/><i><font color="Green" title="E147">A833</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_1_1_2"><i><font color="Green" title="E145">A831</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_11_2_1_1_1_2"/><i><font color="Green" title="E148">A834</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_1_1_2"><i><font color="Green" title="E144">A830</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_11_2_1_1_1_2"/><i><font color="Green" title="E149">A835</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11_2_1_1_1_2"><i><font color="Green" title="E146">A832</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_11_2_1_1_1_2"><i><font color="Green" title="E145">A831</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E150">A836</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_11_2_1_1_1_2"><i><font color="Green" title="E147">A833</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_1_1_2"><i><font color="Green" title="E148">A834</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11_2_1_1_1_2"><i><font color="Green" title="E149">A835</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_11_2_1_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E10:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E151">A837</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E152">A838</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E12:27_11_2_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_1_1_2"><i><font color="Green" title="E148">A834</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E153">A839</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E10:27_11_2_1_1_1_2"><i><font color="Green" title="E151">A837</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E15:27_11_2_1_1_1_2"/><i><font color="Green" title="E154">A840</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_11_2_1_1_1_2"><i><font color="Green" title="E152">A838</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E16:27_11_2_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E17:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E155">A841</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_1_1_2"><i><font color="Green" title="E148">A834</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E18:27_11_2_1_1_1_2"/><i><font color="Green" title="E156">A842</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E19:27_11_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_1_1_2"><i><font color="Green" title="E145">A831</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_11_2_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E21:27_11_2_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E22:27_11_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E157">A843</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_11_2_1_1_1_2"><i><font color="Green" title="E154">A840</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E24:27_11_2_1_1_1_2"/><i><font color="Green" title="E158">A844</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E14:27_11_2_1_1_1_2"><i><font color="Green" title="E153">A839</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E159">A845</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27_11_2_1_1_1_2"><i><font color="Green" title="E150">A836</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E26:27_11_2_1_1_1_2"/><i><font color="Green" title="E160">A846</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E24:27_11_2_1_1_1_2"><i><font color="Green" title="E158">A844</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:27_11_2_1_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E161">A847</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E29:27_11_2_1_1_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E26:27_11_2_1_1_1_2"><i><font color="Green" title="E160">A846</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:27_11_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2"><i><font color="Green" title="E136">A819</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_1"><i><font color="Green" title="E140">A823</font></i></a>, <a class="txt" href="jordan7.html#E9:27_11_2_1_1"><i><font color="Green" title="E143">A826</font></i></a>, <a class="txt" href="jordan7.html#E25:27_11_2_1_1_1_2"><i><font color="Green" title="E159">A845</font></i></a>, <a class="txt" href="jordan7.html#E28:27_11_2_1_1_1_2"><i><font color="Green" title="E161">A847</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E31:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E162">A848</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E32:27_11_2_1_1_1_2"/><i><font color="Green" title="E163">A849</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_2_1_1_1_2"><i><font color="Green" title="E150">A836</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11_2_1_1_1_2"><i><font color="Green" title="E154">A840</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11_2_1_1_1_2"><i><font color="Green" title="E155">A841</font></i></a>, <a class="txt" href="jordan7.html#E18:27_11_2_1_1_1_2"><i><font color="Green" title="E156">A842</font></i></a>, <a class="txt" href="jordan7.html#E24:27_11_2_1_1_1_2"><i><font color="Green" title="E158">A844</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E33:27_11_2_1_1_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E8:27_11_2_1_1_1_2"><i><font color="Green" title="E150">A836</font></i></a>, <a class="txt" href="jordan7.html#E23:27_11_2_1_1_1_2"><i><font color="Green" title="E157">A843</font></i></a>, <a class="txt" href="jordan7.html#E24:27_11_2_1_1_1_2"><i><font color="Green" title="E158">A844</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E34:27_11_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27_11_2_1_1_1_2"><i><font color="Green" title="E162">A848</font></i></a>, <a class="txt" href="jordan7.html#E32:27_11_2_1_1_1_2"><i><font color="Green" title="E163">A849</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_2"><b>suppose </b></a><a NAME="E1:27_11_2_1_2"/><i><font color="Green" title="E137">A850</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E138">A851</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E139">A852</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_11_2_1_2"/><i><font color="Green" title="E140">A853</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_2"><i><font color="Green" title="E137">A850</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E5:27_11_2_1_2"/><i><font color="Green" title="E141">A854</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_11_2_1_2"><i><font color="Green" title="E138">A851</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E142">A855</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11_2_1_2"><i><font color="Green" title="E139">A852</font></i></a>, <a class="txt" href="jordan7.html#E4:27_11_2_1_2"><i><font color="Green" title="E140">A853</font></i></a>, <a class="txt" href="jordan7.html#E5:27_11_2_1_2"><i><font color="Green" title="E141">A854</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_11_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E9:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E143">A856</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E144">A857</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E11:27_11_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E4:27_11_2_1_2"><i><font color="Green" title="E140">A853</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E145">A858</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E9:27_11_2_1_2"><i><font color="Green" title="E143">A856</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E14:27_11_2_1_2"/><i><font color="Green" title="E146">A859</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_11_2_1_2"><i><font color="Green" title="E144">A857</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c13">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E15:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_11_2_1_2"><i><font color="Green" title="E138">A851</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E147">A860</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E17:27_11_2_1_2"/><i><font color="Green" title="E148">A861</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E13:27_11_2_1_2"><i><font color="Green" title="E145">A858</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_11_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_11"><i><font color="Green" title="E118">A727</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E19:27_11_2_1_2"/><b>then </b>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27_11"><i><font color="Green" title="E124">A733</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E20:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E149">A862</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E21:27_11_2_1_2"/><i><font color="Green" title="E150">A863</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_2"><i><font color="Green" title="E137">A850</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E22:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E151">A864</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E23:27_11_2_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E21:27_11_2_1_2"><i><font color="Green" title="E150">A863</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E24:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E152">A865</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E25:27_11_2_1_2"/><i><font color="Green" title="E153">A866</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E12:27_11"><i><font color="Green" title="E129">A738</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2_1_2"><i><font color="Green" title="E137">A850</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E26:27_11_2_1_2"/><i><font color="Green" title="E154">A867</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27_11_2_1_2"><i><font color="Green" title="E151">A864</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E28:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E155">A868</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E24:27_11_2_1_2"><i><font color="Green" title="E152">A865</font></i></a>, <a class="txt" href="jordan7.html#E25:27_11_2_1_2"><i><font color="Green" title="E153">A866</font></i></a>, <a class="txt" href="jordan7.html#E26:27_11_2_1_2"><i><font color="Green" title="E154">A867</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E29:27_11_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E30:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E156">A869</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E31:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E157">A870</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E32:27_11_2_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E25:27_11_2_1_2"><i><font color="Green" title="E153">A866</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E34:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E158">A871</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E30:27_11_2_1_2"><i><font color="Green" title="E156">A869</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E35:27_11_2_1_2"/><i><font color="Green" title="E159">A872</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27_11_2_1_2"><i><font color="Green" title="E157">A870</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c14">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E36:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E22:27_11_2_1_2"><i><font color="Green" title="E151">A864</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E37:27_11_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E38:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_11"><i><font color="Green" title="E130">A739</font></i></a>, <a class="txt" href="jordan7.html#E16:27_11_2_1_2"><i><font color="Green" title="E147">A860</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E39:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E40:27_11_2_1_2"/><b>then </b><i><font color="Green" title="E160">A873</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_11_2_1_2"><i><font color="Green" title="E146">A859</font></i></a>, <a class="txt" href="jordan7.html#E35:27_11_2_1_2"><i><font color="Green" title="E159">A872</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E41:27_11_2_1_2"/><i><font color="Green" title="E161">A874</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E34:27_11_2_1_2"><i><font color="Green" title="E158">A871</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E42:27_11_2_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_11"><i><font color="Green" title="E120">A729</font></i></a>, <a class="txt" href="jordan7.html#E6:27_11"><i><font color="Green" title="E123">A732</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E43:27_11_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>hence </b><a NAME="E44:27_11_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E1:27_11_2"><i><font color="Green" title="E136">A819</font></i></a>, <a class="txt" href="jordan7.html#E7:27_11_2_1_2"><i><font color="Green" title="E142">A855</font></i></a>, <a class="txt" href="jordan7.html#E17:27_11_2_1_2"><i><font color="Green" title="E148">A861</font></i></a>, <a class="txt" href="jordan7.html#E20:27_11_2_1_2"><i><font color="Green" title="E149">A862</font></i></a>, <a class="txt" href="jordan7.html#E28:27_11_2_1_2"><i><font color="Green" title="E155">A868</font></i></a>, <a class="txt" href="jordan7.html#E40:27_11_2_1_2"><i><font color="Green" title="E160">A873</font></i></a>, <a class="txt" href="jordan7.html#E41:27_11_2_1_2"><i><font color="Green" title="E161">A874</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:27_11"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E9:27_11"><i><font color="Green" title="E126">A735</font></i></a>, <a class="txt" href="jordan7.html#E10:27_11"><i><font color="Green" title="E127">A736</font></i></a>, <a class="txt" href="jordan7.html#E15:27_11"><i><font color="Green" title="E132">A741</font></i></a>, <a class="txt" href="jordan7.html#E18:27_11"><i><font color="Green" title="E135">A744</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self" title="JORDAN7:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font> implies  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E129:27"/><i><font color="Green" title="E118">A875</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E130:27"/><i><font color="Green" title="E119">A876</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/>

<a NAME="E131:27"/><i><font color="Green" title="E120">A877</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E132:27"/><i><font color="Green" title="E121">A878</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a></i>;<br/>
<a NAME="E133:27"/><i><font color="Green" title="E122">A879</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E134:27"/><i><font color="Green" title="E123">A880</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E99:27"><i><font color="Green" title="E97">A541</font></i></a>, <a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a></i>;<br/>
<a NAME="E135:27"/><i><font color="Green" title="E124">A881</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="txt" href="jordan7.html#E131:27"><i><font color="Green" title="E120">A877</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E136:27"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E137:27"/><b>then </b><i><font color="Green" title="E125">A882</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green" title="E86">A63</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green" title="E91">A68</font></i></a>, <a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="txt" href="jordan7.html#E135:27"><i><font color="Green" title="E124">A881</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E138:27"/><i><font color="Green" title="E126">A883</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E139:27"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/>
<a NAME="E140:27"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E141:27"/><b>then </b><i><font color="Green" title="E127">A884</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/>
<a NAME="E142:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E143:27"/><b>then </b><i><font color="Green" title="E128">A885</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E141:27"><i><font color="Green" title="E127">A884</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E144:27"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E145:27"/><b>then </b><i><font color="Green" title="E129">A886</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green" title="E79">A56</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E146:27"/><i><font color="Green" title="E130">A887</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E143:27"><i><font color="Green" title="E128">A885</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>
<a NAME="E147:27"/><i><font color="Green" title="E131">A888</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E143:27"><i><font color="Green" title="E128">A885</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E148:27"/><b>then </b><i><font color="Green" title="E132">A889</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E145:27"><i><font color="Green" title="E129">A886</font></i></a>, <a class="txt" href="jordan7.html#E146:27"><i><font color="Green" title="E130">A887</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E149:27"/><i><font color="Green" title="E133">A890</font></i>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_12_1"/>
( <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1_1"><b>suppose </b></a><a NAME="E1:27_12_1_1"/><i><font color="Green" title="E134">A891</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_12_1_1"/><b>then </b><i><font color="Green" title="E135">A892</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_12_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="txt" href="jordan7.html#E1:27_12_1_1"><i><font color="Green" title="E134">A891</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_12_1_1"/><b>then </b><i><font color="Green" title="E136">A893</font></i>: 
<font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E5:27_12_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="txt" href="jordan7.html#E1:27_12_1_1"><i><font color="Green" title="E134">A891</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:27_12_1_1"/><b>then </b><i><font color="Green" title="E137">A894</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_12_1_1"/><i><font color="Green" title="E138">A895</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E138:27"><i><font color="Green" title="E126">A883</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E8:27_12_1_1"/>
<font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E6:27_12_1_1"><i><font color="Green" title="E137">A894</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:27_12_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green" title="E92">A125</font></i></a>, <a class="txt" href="jordan7.html#E148:27"><i><font color="Green" title="E132">A889</font></i></a>, <a class="txt" href="jordan7.html#E2:27_12_1_1"><i><font color="Green" title="E135">A892</font></i></a>, <a class="txt" href="jordan7.html#E4:27_12_1_1"><i><font color="Green" title="E136">A893</font></i></a>, <a class="txt" href="jordan7.html#E7:27_12_1_1"><i><font color="Green" title="E138">A895</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1_2"><b>suppose </b></a><a NAME="E1:27_12_1_2"/><i><font color="Green" title="E134">A896</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font></span><br/><div class="add"><a NAME="E2:27_12_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E3:27_12_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_12_1_2"/><b>then </b><i><font color="Green" title="E135">A897</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E5:27_12_1_2"/><i><font color="Green" title="E136">A898</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E6:27_12_1_2"/><b>then </b><i><font color="Green" title="E137">A899</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E141:27"><i><font color="Green" title="E127">A884</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:27_12_1_2"/><i><font color="Green" title="E138">A900</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_12_1_2"><i><font color="Green" title="E134">A896</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:27_12_1_2"/><i><font color="Green" title="E139">A901</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E9:27_12_1_2"/><i><font color="Green" title="E140">A902</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:27_12_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27_12_1_2"><i><font color="Green" title="E134">A896</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E11:27_12_1_2"/><b>then </b><i><font color="Green" title="E141">A903</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_12_1_2"/><b>then </b><i><font color="Green" title="E142">A904</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><i><font color="Green" title="E143">A905</font></i>: <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E7:27_12_1_2"><i><font color="Green" title="E138">A900</font></i></a>, <a class="txt" href="jordan7.html#E8:27_12_1_2"><i><font color="Green" title="E139">A901</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E9:27_12_1_2"><i><font color="Green" title="E140">A902</font></i></a>, <a class="txt" href="jordan7.html#E11:27_12_1_2"><i><font color="Green" title="E141">A903</font></i></a>, <a class="txt" href="jordan7.html#E12:27_12_1_2"><i><font color="Green" title="E142">A904</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E14:27_12_1_2"/><i><font color="Green" title="E144">A906</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_12_1_2"><i><font color="Green" title="E134">A896</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E15:27_12_1_2"/><i><font color="Green" title="E145">A907</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E141:27"><i><font color="Green" title="E127">A884</font></i></a>, <a class="txt" href="jordan7.html#E5:27_12_1_2"><i><font color="Green" title="E136">A898</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_12_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E17:27_12_1_2"/><b>then </b><i><font color="Green" title="E146">A908</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green" title="E79">A56</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E18:27_12_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E19:27_12_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:27_12_1_2"/><b>then </b><i><font color="Green" title="E147">A909</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E21:27_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_12_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E14:27_12_1_2"><i><font color="Green" title="E144">A906</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E23:27_12_1_2"/><b>then </b><i><font color="Green" title="E148">A910</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E24:27_12_1_2"/><i><font color="Green" title="E149">A911</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27_12_1_2"><i><font color="Green" title="E147">A909</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><b>set </b><font color="Maroon" title="c14">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E25:27_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E26:27_12_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="jordan7.html#E14:27_12_1_2"><i><font color="Green" title="E144">A906</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E27:27_12_1_2"/><b>then </b><i><font color="Green" title="E150">A912</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_12_1_2"><i><font color="Green" title="E135">A897</font></i></a>, <a class="txt" href="jordan7.html#E24:27_12_1_2"><i><font color="Green" title="E149">A911</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E28:27_12_1_2"/><i><font color="Green" title="E151">A913</font></i>: 
<font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E6:27_12_1_2"><i><font color="Green" title="E137">A899</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E29:27_12_1_2"/>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_12_1_2"><i><font color="Green" title="E145">A907</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:27_12_1_2"/><b>then </b><i><font color="Green" title="E152">A914</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E31:27_12_1_2"/><i><font color="Green" title="E153">A915</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E20:27_12_1_2"><i><font color="Green" title="E147">A909</font></i></a>, <a class="txt" href="jordan7.html#E23:27_12_1_2"><i><font color="Green" title="E148">A910</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E32:27_12_1_2"/><b>then </b><i><font color="Green" title="E154">A916</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_12_1_2"><i><font color="Green" title="E143">A905</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E33:27_12_1_2"/><i><font color="Green" title="E155">A917</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27_12_1_2"><i><font color="Green" title="E153">A915</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E34:27_12_1_2"/><b>then </b><i><font color="Green" title="E156">A918</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E35:27_12_1_2"/><i><font color="Green" title="E157">A919</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E33:27_12_1_2"><i><font color="Green" title="E155">A917</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E36:27_12_1_2"/><i><font color="Green" title="E158">A920</font></i>: 
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E15:27_12_1_2"><i><font color="Green" title="E145">A907</font></i></a>, <a class="txt" href="jordan7.html#E27:27_12_1_2"><i><font color="Green" title="E150">A912</font></i></a>, <a class="txt" href="jordan7.html#E31:27_12_1_2"><i><font color="Green" title="E153">A915</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E37:27_12_1_2"/><i><font color="Green" title="E159">A921</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E138:27"><i><font color="Green" title="E126">A883</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E38:27_12_1_2"/>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E33:27_12_1_2"><i><font color="Green" title="E155">A917</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E39:27_12_1_2"/><b>then </b><i><font color="Green" title="E160">A922</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E13:27_12_1_2"><i><font color="Green" title="E143">A905</font></i></a>, <a class="txt" href="jordan7.html#E31:27_12_1_2"><i><font color="Green" title="E153">A915</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E40:27_12_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="jordan7.html#E145:27"><i><font color="Green" title="E129">A886</font></i></a>, <a class="txt" href="jordan7.html#E146:27"><i><font color="Green" title="E130">A887</font></i></a>, <a class="txt" href="jordan7.html#E147:27"><i><font color="Green" title="E131">A888</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:27_12_1_2"/><b>then </b><i><font color="Green" title="E161">A923</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green" title="E77">A54</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E42:27_12_1_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>, <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>, <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="jordan7.html#E15:27"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E17:27_12_1_2"><i><font color="Green" title="E146">A908</font></i></a>, <a class="txt" href="jordan7.html#E28:27_12_1_2"><i><font color="Green" title="E151">A913</font></i></a>, <a class="txt" href="jordan7.html#E30:27_12_1_2"><i><font color="Green" title="E152">A914</font></i></a>, <a class="txt" href="jordan7.html#E32:27_12_1_2"><i><font color="Green" title="E154">A916</font></i></a>, <a class="txt" href="jordan7.html#E34:27_12_1_2"><i><font color="Green" title="E156">A918</font></i></a>, <a class="txt" href="jordan7.html#E35:27_12_1_2"><i><font color="Green" title="E157">A919</font></i></a>, <a class="txt" href="jordan7.html#E36:27_12_1_2"><i><font color="Green" title="E158">A920</font></i></a>, <a class="txt" href="jordan7.html#E37:27_12_1_2"><i><font color="Green" title="E159">A921</font></i></a>, <a class="ref" href="jordan7.html#T19" target="_self" title="JORDAN7:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E43:27_12_1_2"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green" title="E92">A125</font></i></a>, <a class="txt" href="jordan7.html#E37:27_12_1_2"><i><font color="Green" title="E159">A921</font></i></a>, <a class="txt" href="jordan7.html#E39:27_12_1_2"><i><font color="Green" title="E160">A922</font></i></a>, <a class="txt" href="jordan7.html#E41:27_12_1_2"><i><font color="Green" title="E161">A923</font></i></a>, <a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_13"/><i><font color="Green" title="E134">A924</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_13_1"/>
( <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> or <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1_1"><b>suppose </b></a><a NAME="E1:27_13_1_1"/><i><font color="Green" title="E135">A925</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_13_1_1"/>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_13_1_1"/><b>then </b><i><font color="Green" title="E136">A926</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:27_13_1_1"/><i><font color="Green" title="E137">A927</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="txt" href="jordan7.html#E1:27_13_1_1"><i><font color="Green" title="E135">A925</font></i></a>, <a class="ref" href="finseq_1.html#T85" title="FINSEQ_1:th.85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:27_13_1_1"/><i><font color="Green" title="E138">A928</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E133:27"><i><font color="Green" title="E122">A879</font></i></a>, <a class="txt" href="jordan7.html#E1:27_13_1_1"><i><font color="Green" title="E135">A925</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:27_13_1_1"/><b>then </b><i><font color="Green" title="E139">A929</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27_13_1_1"><i><font color="Green" title="E137">A927</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E7:27_13_1_1"/>
<font color="Maroon" title="c4">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27_13_1_1"><i><font color="Green" title="E138">A928</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_13_1_1"/><b>then </b><i><font color="Green" title="E140">A930</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="jordan7.html#E11:27"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="jordan7.html#E6:27_13_1_1"><i><font color="Green" title="E139">A929</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_13_1_1"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E10:27_13_1_1"/><b>then </b><i><font color="Green" title="E141">A931</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green" title="E79">A56</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E11:27_13_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E12:27_13_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_13_1_1"/><b>then </b><i><font color="Green" title="E142">A932</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E14:27_13_1_1"/><i><font color="Green" title="E143">A933</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E15:27_13_1_1"/><i><font color="Green" title="E144">A934</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E13:27_13_1_1"><i><font color="Green" title="E142">A932</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c13">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E16:27_13_1_1"/><i><font color="Green" title="E145">A935</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:27_13_1_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E18:27_13_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:27_13_1_1"/><b>then </b><i><font color="Green" title="E146">A936</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_13_1_1"><i><font color="Green" title="E144">A934</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E20:27_13_1_1"/><i><font color="Green" title="E147">A937</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E13:27_13_1_1"><i><font color="Green" title="E142">A932</font></i></a>, <a class="txt" href="jordan7.html#E14:27_13_1_1"><i><font color="Green" title="E143">A933</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:27_13_1_1"/><b>then </b><i><font color="Green" title="E148">A938</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E10:27_13_1_1"><i><font color="Green" title="E141">A931</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E22:27_13_1_1"/>
<font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E20:27_13_1_1"><i><font color="Green" title="E147">A937</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E23:27_13_1_1"/><b>then </b>
<font color="Maroon" title="c5">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E18:27"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E24:27_13_1_1"/><b>then </b>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E16:27"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E1:27_13"><i><font color="Green" title="E134">A924</font></i></a>, <a class="txt" href="jordan7.html#E3:27_13_1_1"><i><font color="Green" title="E136">A926</font></i></a>, <a class="txt" href="jordan7.html#E8:27_13_1_1"><i><font color="Green" title="E140">A930</font></i></a>, <a class="txt" href="jordan7.html#E21:27_13_1_1"><i><font color="Green" title="E148">A938</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E25:27_13_1_1"/><b>then </b><i><font color="Green" title="E149">A939</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="jordan6.html#D9" title="JORDAN6:def.9">JORDAN6:def 9</a></i>;<br/><a NAME="E26:27_13_1_1"/><i><font color="Green" title="E150">A940</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K25" title="PSCOMP_1:func.25">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green" title="E54">A31</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E10:27_13_1_1"><i><font color="Green" title="E141">A931</font></i></a>, <a class="txt" href="jordan7.html#E16:27_13_1_1"><i><font color="Green" title="E145">A935</font></i></a>, <a class="txt" href="jordan7.html#E20:27_13_1_1"><i><font color="Green" title="E147">A937</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E27:27_13_1_1"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E10:27_13_1_1"><i><font color="Green" title="E141">A931</font></i></a>, <a class="txt" href="jordan7.html#E19:27_13_1_1"><i><font color="Green" title="E146">A936</font></i></a>, <a class="txt" href="jordan7.html#E20:27_13_1_1"><i><font color="Green" title="E147">A937</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E28:27_13_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27_13_1_1"><i><font color="Green" title="E149">A939</font></i></a>, <a class="txt" href="jordan7.html#E26:27_13_1_1"><i><font color="Green" title="E150">A940</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1_2"><b>suppose </b></a><a NAME="E1:27_13_1_2"/><i><font color="Green" title="E135">A941</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:27_13_1_2"/><b>then </b><i><font color="Green" title="E136">A942</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:27_13_1_2"/><i><font color="Green" title="E137">A943</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E4:27_13_1_2"/><i><font color="Green" title="E138">A944</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="txt" href="jordan7.html#E2:27_13_1_2"><i><font color="Green" title="E136">A942</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_13_1_2"/><i><font color="Green" title="E139">A945</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E1:27_13_1_2"><i><font color="Green" title="E135">A941</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:27_13_1_2"/><i><font color="Green" title="E140">A946</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E3:27_13_1_2"><i><font color="Green" title="E137">A943</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27_13_1_2"><i><font color="Green" title="E136">A942</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_13_1_2"/><b>then </b><i><font color="Green" title="E141">A947</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E60:27"><i><font color="Green" title="E68">A45</font></i></a>, <a class="txt" href="jordan7.html#E4:27_13_1_2"><i><font color="Green" title="E138">A944</font></i></a>, <a class="txt" href="jordan7.html#E5:27_13_1_2"><i><font color="Green" title="E139">A945</font></i></a>, <a class="txt" href="jordan7.html#E6:27_13_1_2"><i><font color="Green" title="E140">A946</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_13_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E10:27_13_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_13_1_2"/><b>then </b><i><font color="Green" title="E142">A948</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E12:27_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">h1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E5:27_13_1_2"><i><font color="Green" title="E139">A945</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_13_1_2"/><b>then </b><i><font color="Green" title="E143">A949</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E15:27_13_1_2"/><i><font color="Green" title="E144">A950</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E11:27_13_1_2"><i><font color="Green" title="E142">A948</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c13">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E16:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E2:27_13_1_2"><i><font color="Green" title="E136">A942</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E17:27_13_1_2"/><b>then </b><i><font color="Green" title="E145">A951</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E18:27_13_1_2"/><i><font color="Green" title="E146">A952</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E11:27_13_1_2"><i><font color="Green" title="E142">A948</font></i></a>, <a class="txt" href="jordan7.html#E14:27_13_1_2"><i><font color="Green" title="E143">A949</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:27_13_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E129:27"><i><font color="Green" title="E118">A875</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E20:27_13_1_2"/><b>then </b>
<font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:27_13_1_2"/><b>then </b><i><font color="Green" title="E147">A953</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E22:27_13_1_2"/>
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c8">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green" title="E74">A51</font></i></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></i>;<br/><a NAME="E23:27_13_1_2"/><b>then </b><i><font color="Green" title="E148">A954</font></i>: 
<font color="Maroon" title="c9">h0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h21</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E57:27"><i><font color="Green" title="E66">A43</font></i></a>, <a class="txt" href="jordan7.html#E59:27"><i><font color="Green" title="E67">A44</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E24:27_13_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E25:27_13_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E26:27_13_1_2"/><b>then </b><i><font color="Green" title="E149">A955</font></i>: 
2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T60" title="BINARITH:th.60">BINARITH:60</a></i>;<br/><a NAME="E27:27_13_1_2"/><i><font color="Green" title="E150">A956</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/><a NAME="E28:27_13_1_2"/><i><font color="Green" title="E151">A957</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27_13_1_2"><i><font color="Green" title="E149">A955</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><b>set </b><font color="Maroon" title="c14">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/><a NAME="E29:27_13_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green" title="E63">A40</font></i></a>, <a class="txt" href="jordan7.html#E54:27"><i><font color="Green" title="E65">A42</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green" title="E69">A46</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green" title="E70">A47</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green" title="E73">A50</font></i></a>, <a class="ref" href="binarith.html#T57" title="BINARITH:th.57">BINARITH:57</a></i>;<br/><a NAME="E30:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E130:27"><i><font color="Green" title="E119">A876</font></i></a>, <a class="txt" href="jordan7.html#E17:27_13_1_2"><i><font color="Green" title="E145">A951</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E31:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E32:27_13_1_2"/><b>then </b><i><font color="Green" title="E152">A958</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E15:27_13_1_2"><i><font color="Green" title="E144">A950</font></i></a>, <a class="txt" href="jordan7.html#E28:27_13_1_2"><i><font color="Green" title="E151">A957</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E26:27_13_1_2"><i><font color="Green" title="E149">A955</font></i></a>, <a class="txt" href="jordan7.html#E27:27_13_1_2"><i><font color="Green" title="E150">A956</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E34:27_13_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green" title="E78">A55</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green" title="E90">A67</font></i></a>, <a class="txt" href="jordan7.html#E1:27_13"><i><font color="Green" title="E134">A924</font></i></a>, <a class="txt" href="jordan7.html#E8:27_13_1_2"><i><font color="Green" title="E141">A947</font></i></a>, <a class="txt" href="jordan7.html#E18:27_13_1_2"><i><font color="Green" title="E146">A952</font></i></a>, <a class="txt" href="jordan7.html#E21:27_13_1_2"><i><font color="Green" title="E147">A953</font></i></a>, <a class="txt" href="jordan7.html#E23:27_13_1_2"><i><font color="Green" title="E148">A954</font></i></a>, <a class="txt" href="jordan7.html#E32:27_13_1_2"><i><font color="Green" title="E152">A958</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E151:27"/>
 <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">h0</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green" title="E87">A64</font></i></a>, <a class="txt" href="jordan7.html#E132:27"><i><font color="Green" title="E121">A878</font></i></a>, <a class="txt" href="jordan7.html#E134:27"><i><font color="Green" title="E123">A880</font></i></a>, <a class="txt" href="jordan7.html#E137:27"><i><font color="Green" title="E125">A882</font></i></a>, <a class="txt" href="jordan7.html#E149:27"><i><font color="Green" title="E133">A890</font></i></a>, <a class="ref" href="jordan7.html#T14" target="_self" title="JORDAN7:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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