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<span class="kw">let </span><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b3">A</font> being  <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> holds <br/>( ( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b3">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> ) &amp; ( for <font color="Olive" title="b4">n</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="b4">n</font> &amp; <font color="Olive" title="b4">n</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="b3">A</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b3">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/> for <font color="Olive" title="b2">A</font> being  <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> holds <br/>( ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> ) &amp; ( for <font color="Olive" title="b3">n</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">n</font> &amp; <font color="Olive" title="b3">n</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">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b2">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">D</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c4">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> holds <br/>( ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> ) &amp; ( for <font color="Olive" title="b2">n</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">n</font> &amp; <font color="Olive" title="b2">n</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">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> implies ( ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</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">n</font> &amp; <font color="Olive" title="b1">n</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="c5">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">fd</font> =  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">mf</font> =  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">h</font> =  <a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">fh</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E1:26"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:26"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span> <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="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D6" title="RFUNCT_3:def.6">RFUNCT_3:def 6</a></span>;<br/>
<a NAME="E4:26"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/>
<a NAME="E5:26"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p5">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:26"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
 <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</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="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span>,<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="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:26"/>
( <font color="Maroon" title="c4">F</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</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">n</font> &amp; <font color="Olive" title="b1">n</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="c5">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E8:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rearran1.html#T12" target="_self" title="REARRAN1:th.12">Th12</a></span>;<br/>
<a NAME="E9:26"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/>
<a NAME="E10:26"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:26"/>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</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">n</font> &amp; <font color="Olive" title="b1">n</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="c5">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span><br/>

<a NAME="E2:26_1"/><span class="lab"><font color="Green" title="E32">A11</font></span>: 
for <font color="Olive" title="b1">n</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="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/>

<a NAME="E2:26_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E3:26_1_1"/>
<font color="Maroon" title="c10">n</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="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:26_1_1"/><span class="kw">then </span>
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="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:26_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E6:26_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="rearran1.html#T2" target="_self" title="REARRAN1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:26_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">m</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">r1</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">assume </span><a NAME="E3:26_1_2"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E4:26_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:26_1_2"/><span class="lab"><font color="Green" title="E36">A17</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="rfunct_3.html#D6" title="RFUNCT_3:def.6">RFUNCT_3:def 6</a></span>;<br/><a NAME="E6:26_1_2"/><span class="lab"><font color="Green" title="E37">A18</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a></span>;<br/><a NAME="E7:26_1_2"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:26_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A19</font></span>: 
<font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_2_1_1"/>
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:26_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A20</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span><br/><div class="add"><a NAME="E2:26_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/><a NAME="E3:26_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E5:26_1_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/><a NAME="E4:26_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T77" title="RFUNCT_1:th.77">RFUNCT_1:77</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> = 
<font color="Maroon" title="c12">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A21</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c12">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="rfunct_1.html#T79" title="RFUNCT_1:th.79">RFUNCT_1:79</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A20</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:26_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A22</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">m1</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c13">r2</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:26_1_2_1_1_2"/>
<font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_2"><span class="lab"><font color="Green" title="E38">A19</font></span></a>, <a class="txt" href="#E1:26_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:26_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:26_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A23</font></span>: 
<font color="Maroon" title="c10">m</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:26_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/><a NAME="E7:26_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E5:26_1_2"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/><a NAME="E8:26_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A24</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</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">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_2"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T77" title="RFUNCT_1:th.77">RFUNCT_1:77</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">r2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_2"><span class="lab"><font color="Green" title="E37">A18</font></span></a>, <a class="ref" href="rfunct_1.html#T79" title="RFUNCT_1:th.79">RFUNCT_1:79</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_2"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E5:26_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:26_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:26_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c10">n</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">n</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="c5">A</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:26"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">and </span><br/><a NAME="E13:26"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
<font color="Maroon" title="c10">n</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="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E14:26"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span><br/>

<a NAME="E15:26"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<a NAME="E16:26"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> holds <br/> not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> implies  not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_2"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<span class="kw">assume </span><a NAME="E2:26_2"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:26_2"/>
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:26_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="rearran1.html#T2" target="_self" title="REARRAN1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:26_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E2:26_2"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:26"/><span class="lab"><font color="Green" title="E36">A32</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:26"><span class="lab"><font color="Green" title="E33">A27</font></span></a></span>;<br/>
<a NAME="E18:26"/><span class="lab"><font color="Green" title="E37">A33</font></span>: 
<font color="Maroon" title="c10">n</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="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E19:26"/><span class="lab"><font color="Green" title="E38">A34</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E31">A25</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E20:26"/><span class="lab"><font color="Green" title="E39">A35</font></span>: 
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="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c10">n</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="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c10">n</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="c11">k</font> implies <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3"/><span class="lab"><font color="Green" title="E40">A36</font></span>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c10">n</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="c11">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font></span><br/>

<a NAME="E2:26_3"/>
<font color="Maroon" title="c10">n</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="txt" href="#E19:26"><span class="lab"><font color="Green" title="E38">A34</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E3:26_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3"><span class="lab"><font color="Green" title="E40">A36</font></span></a>, <a class="ref" href="rearran1.html#T2" target="_self" title="REARRAN1:th.2">Th2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:26_3"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E36">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c11">fdn</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">mfn</font> =  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">n0</font> = <font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<a NAME="E21:26"/><span class="lab"><font color="Green" title="E40">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/>
<a NAME="E22:26"/><span class="lab"><font color="Green" title="E41">A38</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E23:26"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</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> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E42">A39</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c10">n</font> <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> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>
;<br/>
<a NAME="E25:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A40</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E26:26"/><span class="lab"><font color="Green" title="E44">A41</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c15">m1</font> = <font color="Maroon" title="c14">m</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">r1</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><span class="kw">assume </span><a NAME="E3:26_5"/><span class="lab"><font color="Green" title="E45">A42</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/><a NAME="E4:26_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:26_5"/>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E6:26_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A44</font></span>: 
<font color="Maroon" title="c14">m</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:26_5"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="rfunct_3.html#D6" title="RFUNCT_3:def.6">RFUNCT_3:def 6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_5_1_1"/>
( <font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1_1"><span class="kw">case </span></a><a NAME="E1:26_5_1_1_1"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c17">r2</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:26_5_1_1_1"/>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_5_1_1_1"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</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="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:26"><span class="lab"><font color="Green" title="E37">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:26_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:26_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E7:26_5_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/><a NAME="E8:26_5_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E7:26_5"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/><a NAME="E9:26_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A47</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/><a NAME="E10:26_5_1_1_1"/>
not <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E16:26"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E1:26_5_1_1_1"><span class="lab"><font color="Green" title="E49">A46</font></span></a></span>;<br/><a NAME="E11:26_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A48</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T80" title="RFUNCT_1:th.80">RFUNCT_1:80</a></span>;<br/><a NAME="E12:26_5_1_1_1"/><span class="lab"><font color="Green" title="E52">A49</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E1:26_5_1_1_1"><span class="lab"><font color="Green" title="E49">A46</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E13:26_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A50</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T77" title="RFUNCT_1:th.77">RFUNCT_1:77</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">r2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_5_1_1_1"><span class="lab"><font color="Green" title="E50">A47</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="txt" href="#E26:26"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="txt" href="#E11:26_5_1_1_1"><span class="lab"><font color="Green" title="E51">A48</font></span></a>, <a class="txt" href="#E12:26_5_1_1_1"><span class="lab"><font color="Green" title="E52">A49</font></span></a>, <a class="txt" href="#E13:26_5_1_1_1"><span class="lab"><font color="Green" title="E53">A50</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1_2"><span class="kw">case </span></a><a NAME="E1:26_5_1_1_2"/><span class="lab"><font color="Green" title="E49">A51</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/><div class="add"><a NAME="E2:26_5_1_1_2"/><span class="kw">then </span>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T4" title="FINSEQ_2:th.4">FINSEQ_2:4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">mn</font> = <font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T5" title="FINSEQ_2:th.5">FINSEQ_2:5</a></span>;<br/><a NAME="E4:26_5_1_1_2"/><span class="lab"><font color="Green" title="E50">A52</font></span>: 
<font color="Maroon" title="c10">n</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="c14">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_5_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:26_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E39">A35</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a></span>;<br/><a NAME="E6:26_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A53</font></span>: 
<span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T79" title="RFUNCT_1:th.79">RFUNCT_1:79</a></span>;<br/><a NAME="E7:26_5_1_1_2"/><span class="lab"><font color="Green" title="E52">A54</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">mn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5_1_1_2"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_5_1_1_2_1_1"/>
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> or <font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:26_5_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E53">A55</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/><div class="add"><a NAME="E2:26_5_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/><a NAME="E3:26_5_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E7:26_5"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/><a NAME="E4:26_5_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A56</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/><a NAME="E5:26_5_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A57</font></span>: 
<font color="Maroon" title="c17">mn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E7:26_5_1_1_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="txt" href="#E1:26_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="lab"><font color="Green" title="E56">A58</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> = 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E42">A39</font></span></a>, <a class="txt" href="#E6:26_5"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E1:26_5_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">mn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T28" title="RFINSEQ:th.28">RFINSEQ:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">mn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E5:26_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E55">A57</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c16">r1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:26_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E53">A55</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>
;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T77" title="RFUNCT_1:th.77">RFUNCT_1:77</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> = 
<font color="Maroon" title="c16">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E54">A56</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_5_1_1_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E6:26_5_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A58</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_5_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:26_5_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A59</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c18">r2</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E3:26_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">m</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_5"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E1:26_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A59</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:26_5_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:26_5_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A60</font></span>: 
<font color="Maroon" title="c14">m</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> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E6:26_5_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>;<br/><a NAME="E7:26_5_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="txt" href="#E7:26_5"><span class="lab"><font color="Green" title="E48">A45</font></span></a>, <a class="ref" href="rfunct_3.html#D7" title="RFUNCT_3:def.7">RFUNCT_3:def 7</a></span>;<br/><a NAME="E8:26_5_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A61</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E25:26"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E3:26_5"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="rfunct_3.html#D8" title="RFUNCT_3:def.8">RFUNCT_3:def 8</a></span>;<br/><a NAME="E9:26_5_1_1_2_1_1_2"/>
<font color="Maroon" title="c17">mn</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E6:26_5"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E10:26_5_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A62</font></span>: 
<font color="Maroon" title="c17">mn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_5_1_1_2"><span class="lab"><font color="Green" title="E52">A54</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><span class="lab"><font color="Green" title="E57">A63</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> = 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E42">A39</font></span></a>, <a class="txt" href="#E6:26_5"><span class="lab"><font color="Green" title="E47">A44</font></span></a>, <a class="txt" href="#E1:26_5_1_1_2"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">mn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#T28" title="RFINSEQ:th.28">RFINSEQ:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">mn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E10:26_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_5"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E5:26_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A60</font></span></a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a></span>
;<br/> <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font></span>)</span> <a href="rearran1.html#K1" title="REARRAN1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T77" title="RFUNCT_1:th.77">RFUNCT_1:77</a></span>
;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">r1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r2</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_5_1_1_2"><span class="lab"><font color="Green" title="E51">A53</font></span></a>, <a class="txt" href="#E11:26_5_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A63</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:26_5_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:26_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</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="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E28:26"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E24:26"><span class="lab"><font color="Green" title="E42">A39</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
