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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><br/>  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Olive" title="b3">i</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="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">i</font>) &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b3">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> holds <br/>( <font color="Olive" title="b2">h</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b2">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b3">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">h</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Olive" title="b1">i</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="c2">f</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">i</font>) &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> holds <br/>( <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c4">i</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="c2">f</font> &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>) &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> implies ( <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<font color="Maroon" title="c4">i</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="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E4:16"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E5:16"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">and </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">and </span><br/><a NAME="E7:16"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/>

<a NAME="E8:16"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font>;<br/>
<a NAME="E9:16"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<a NAME="E10:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</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="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E11:16"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
 <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="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">h</font>
 
<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="E12:16"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E13:16"/><span class="lab"><font color="Green" title="E26">A13</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<a NAME="E14:16"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">q2</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E15:16"/><span class="lab"><font color="Green" title="E28">A15</font></span>: 
 <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="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<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="E16:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E17:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E18:16"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E19:16"/><span class="lab"><font color="Green" title="E32">A19</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E20:16"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E21:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A20</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</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="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="ref" href="partfun2.html#T10" title="PARTFUN2:th.10">PARTFUN2:10</a></span>;<br/>
<a NAME="E22:16"/><span class="lab"><font color="Green" title="E34">A21</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" title="RLTOPSP1:th.68">RLTOPSP1:68</a></span>;<br/>
<a NAME="E23:16"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<a NAME="E24:16"/><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c4">i</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="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E25:16"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E26:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A24</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="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</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> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E27:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A25</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="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 
<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">then </span><span class="lab"><font color="Green" title="E39">A26</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> = 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_1.html#T22" title="FINSEQ_1:th.22">FINSEQ_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>
;<br/>
<a NAME="E29:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/>
<a NAME="E30:16"/><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<font color="Maroon" title="c4">i</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E42">A29</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:16"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/>
<a NAME="E32:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A30</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E33:16"/><span class="lab"><font color="Green" title="E44">A31</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">thus </span><a NAME="E34:16"/><span class="lab"><font color="Green" title="E45">A32</font></span>: 
<font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V4" title="TOPREAL1:attr.4">being_S-Seq</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">Z</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">m</font> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c3">h</font> ) </span> } </span> ;<br/><span class="kw">given </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E2:16_3_1"/><span class="lab"><font color="Green" title="E46">A33</font></span>: 
<font color="Maroon" title="c11">x</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="c3">h</font>
 <span class="kw">and </span><br/><a NAME="E3:16_3_1"/><span class="lab"><font color="Green" title="E47">A34</font></span>: 
<font color="Maroon" title="c12">y</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="c3">h</font>
 <span class="kw">and </span><br/><a NAME="E4:16_3_1"/><span class="lab"><font color="Green" title="E48">A35</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">and </span><br/><a NAME="E5:16_3_1"/><span class="lab"><font color="Green" title="E49">A36</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:16_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">m</font> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c3">h</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E2:16_3_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">k1</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> <span class="kw"> such that </span><br/><a NAME="E8:16_3_1"/><span class="lab"><font color="Green" title="E50">A37</font></span>: 
<font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E9:16_3_1"/><span class="lab"><font color="Green" title="E51">A38</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k1</font>
 <span class="kw">and </span><br/><a NAME="E10:16_3_1"/><span class="lab"><font color="Green" title="E52">A39</font></span>: 
<font color="Maroon" title="c13">k1</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="c3">h</font>
 ;<br/><a NAME="E11:16_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">m</font> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">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> <font color="Maroon" title="c3">h</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="txt" href="#E3:16_3_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k2</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> <span class="kw"> such that </span><br/><a NAME="E13:16_3_1"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y</font>
 <span class="kw">and </span><br/><a NAME="E14:16_3_1"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k2</font>
 <span class="kw">and </span><br/><a NAME="E15:16_3_1"/><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c14">k2</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="c3">h</font>
 ;<br/><span class="lab"><font color="Green" title="E56">A43</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> = 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1"><span class="lab"><font color="Green" title="E46">A33</font></span></a>, <a class="txt" href="#E4:16_3_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E8:16_3_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3_1"><span class="lab"><font color="Green" title="E47">A34</font></span></a>, <a class="txt" href="#E13:16_3_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/><a NAME="E17:16_3_1"/>
<font color="Maroon" title="c14">k2</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E15:16_3_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:16_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A44</font></span>: 
<font color="Maroon" title="c14">k2</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E19:16_3_1"/>
<font color="Maroon" title="c13">k1</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E10:16_3_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E20:16_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: 
<font color="Maroon" title="c13">k1</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E21:16_3_1"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E22:16_3_1"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_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="E1:16_3_1_2_1"/>
( ( <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E10:16_3_1"><span class="lab"><font color="Green" title="E52">A39</font></span></a>, <a class="txt" href="#E15:16_3_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_1"/>
( <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_3_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_3_1"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E8:16_3_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E13:16_3_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_2"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
( <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<font color="Maroon" title="c14">k2</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="c13">k1</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_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="E1:16_3_1_2_1_2_1_1"/>
( <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k1</font> or <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1_2_1_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">k1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_3_1_2_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">k1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A50</font></span>: 
<font color="Maroon" title="c14">k2</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="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_2_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="E1:16_3_1_2_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A50</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A51</font></span>: 
<font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A52</font></span>: 
<font color="Maroon" title="c14">k2</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A53</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A52</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/><a NAME="E5:16_3_1_2_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">k2</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A54</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c14">k2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><a NAME="E7:16_3_1_2_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c14">k2</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E22:16_3_1"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/><a NAME="E8:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E6:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E4:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:16_3_1_2_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A55</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:16_3_1_2_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:16_3_1_2_1_2_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E18:16_3_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="txt" href="#E9:16_3_1_2_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="partfun2.html#T10" title="PARTFUN2:th.10">PARTFUN2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A56</font></span>: 
<font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A57</font></span>: 
<font color="Maroon" title="c14">k2</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A58</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c14">k2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><a NAME="E4:16_3_1_2_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A56</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A59</font></span>: 
<font color="Maroon" title="c14">k2</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A59</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/><a NAME="E8:16_3_1_2_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c14">k2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a>, <a class="txt" href="#E2:16_3_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_3_1_2_1_2_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E3:16_3_1_2_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_1_2_1_2_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_1_2_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_3"/><span class="lab"><font color="Green" title="E61">A60</font></span>: 
( <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A61</font></span>: 
<font color="Maroon" title="c13">k1</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">k2</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_3_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="E1:16_3_1_2_1_3_1_1"/>
( <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k2</font> or <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1_2_1_3"><span class="lab"><font color="Green" title="E62">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_3_1_1_1"/>
<font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:16_3_1_2_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E31:16"><span class="lab"><font color="Green" title="E42">A29</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_3_1_1_2"/>
<font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A62</font></span>: 
<font color="Maroon" title="c13">k1</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="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_3"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1_2_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_3_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="E1:16_3_1_2_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> or <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1_2_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_2_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A63</font></span>: 
<font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A64</font></span>: 
<font color="Maroon" title="c13">k1</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A65</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A64</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/><a NAME="E5:16_3_1_2_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c13">k1</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A66</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><a NAME="E7:16_3_1_2_1_3_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">k1</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E21:16_3_1"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/><a NAME="E8:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="txt" href="#E6:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A66</font></span></a>, <a class="txt" href="#E4:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A65</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:16_3_1_2_1_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A67</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:16_3_1_2_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:16_3_1_2_1_3_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E20:16_3_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E9:16_3_1_2_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="partfun2.html#T10" title="PARTFUN2:th.10">PARTFUN2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A68</font></span>: 
<font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A69</font></span>: 
<font color="Maroon" title="c13">k1</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:16"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A70</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">k1</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><a NAME="E4:16_3_1_2_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A71</font></span>: 
<font color="Maroon" title="c13">k1</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_1_2_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/><a NAME="E8:16_3_1_2_1_3_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">k1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E29:16"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="txt" href="#E16:16_3_1"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="txt" href="#E1:16_3_1_2_1_3"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="txt" href="#E2:16_3_1_2_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A69</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_3_1_2_1_3_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E3:16_3_1_2_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_1_2_1_3_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_1_2_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:16_3_1_2_1_4"/><span class="lab"><font color="Green" title="E61">A72</font></span>: 
( <font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:16_3_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</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="E3:16_3_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A73</font></span>: 
<font color="Maroon" title="c14">k2</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E14:16_3_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:16_3_1_2_1_4"/>
<font color="Maroon" title="c13">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1_2_1_4"><span class="lab"><font color="Green" title="E61">A72</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:16_3_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A74</font></span>: 
<font color="Maroon" title="c13">k1</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E9:16_3_1"><span class="lab"><font color="Green" title="E51">A38</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E5:16_3_1_2_1_4"><span class="lab"><font color="Green" title="E63">A74</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3_1"><span class="lab"><font color="Green" title="E48">A35</font></span></a>, <a class="txt" href="#E8:16_3_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E13:16_3_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E3:16_3_1_2_1_4"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E3:16_3_1_2_1_4"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:16_3_1_2_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E5:16_3_1"><span class="lab"><font color="Green" title="E49">A36</font></span></a>, <a class="txt" href="#E8:16_3_1"><span class="lab"><font color="Green" title="E50">A37</font></span></a>, <a class="txt" href="#E13:16_3_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="txt" href="#E20:16_3_1"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="txt" href="#E18:16_3_1"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E24:16_3_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:16_3"/>
<font color="Maroon" title="c3">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D8" title="TOPREAL1:def.8">TOPREAL1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E3:16_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E33:16"><span class="lab"><font color="Green" title="E44">A31</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:16_3"/>
<font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> or  not <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16_3_2"/><span class="lab"><font color="Green" title="E46">A75</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">and </span><br/><a NAME="E2:16_3_2"/><span class="lab"><font color="Green" title="E47">A76</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:16_3_2"/><span class="lab"><font color="Green" title="E48">A77</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E4:16_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E5:16_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</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> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E6:16_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A78</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E3:16_3_2"><span class="lab"><font color="Green" title="E48">A77</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E50">A79</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_2"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/>
<a NAME="E8:16_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E9:16_3_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E10:16_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E11:16_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E12:16_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A80</font></span>: 
<font color="Maroon" title="c10">j</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E1:16_3_2"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E52">A81</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>) = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E6:16_3_2"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16_3_2"><span class="lab"><font color="Green" title="E51">A80</font></span></a>, <a class="txt" href="#E6:16_3_2"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/>
<a NAME="E14:16_3_2"/>
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E15:16_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A82</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_2"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E16:16_3_2"/><span class="lab"><font color="Green" title="E54">A83</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E17:16_3_2"/>
<font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E16">A3</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="E18:16_3_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</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="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E19:16_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A84</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E20:16_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A85</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E1:16_3_2"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_2_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_2_3_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="E1:16_3_2_3_1"/>
( <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_2_3_1_1"/><span class="lab"><font color="Green" title="E57">A86</font></span>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:16_3_2_3_1_1"/>
<font color="Maroon" title="c10">j</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:16_3_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</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="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:16_3_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A87</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_2"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><a NAME="E5:16_3_2_3_1_1"/>
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_2"><span class="lab"><font color="Green" title="E47">A76</font></span></a>, <a class="txt" href="#E3:16_3_2"><span class="lab"><font color="Green" title="E48">A77</font></span></a>, <a class="txt" href="#E16:16_3_2"><span class="lab"><font color="Green" title="E54">A83</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><a NAME="E6:16_3_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_3_2_3_1_1"><span class="lab"><font color="Green" title="E58">A87</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:16_3_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A88</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E8:16_3_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E20:16_3_2"><span class="lab"><font color="Green" title="E56">A85</font></span></a>, <a class="txt" href="#E13:16_3_2"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="txt" href="#E7:16_3_2"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="txt" href="#E1:16_3_2_3_1_1"><span class="lab"><font color="Green" title="E57">A86</font></span></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_3_2_3_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_3_2_3_1_1"><span class="lab"><font color="Green" title="E59">A88</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_2_3_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:16_3_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">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="E3:16_3_2_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A89</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E15:16_3_2"><span class="lab"><font color="Green" title="E53">A82</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E6:16_3_2"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_2"><span class="lab"><font color="Green" title="E49">A78</font></span></a>, <a class="txt" href="#E3:16_3_2_3_1_2"><span class="lab"><font color="Green" title="E57">A89</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:16_3_2_3_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:16"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="txt" href="#E1:16_3_2"><span class="lab"><font color="Green" title="E46">A75</font></span></a>, <a class="txt" href="#E19:16_3_2"><span class="lab"><font color="Green" title="E55">A84</font></span></a>, <a class="txt" href="#E13:16_3_2"><span class="lab"><font color="Green" title="E52">A81</font></span></a>, <a class="txt" href="#E7:16_3_2"><span class="lab"><font color="Green" title="E50">A79</font></span></a>, <a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E22:16_3_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E5:16_3"/>
<font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">h</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font>, <font color="Maroon" title="c11">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">m</font> <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#K1" title="NAT_1:func.1">+</a> 1 or  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_3_3"/><span class="lab"><font color="Green" title="E46">A90</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span><br/>

<a NAME="E2:16_3_3"/>
<font color="Maroon" title="c10">n</font> <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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E3:16_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A91</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E4:16_3_3"/><span class="lab"><font color="Green" title="E48">A92</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#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E5:16_3_3"/><span class="lab"><font color="Green" title="E49">A93</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_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="E1:16_3_3_1_1"/>
( <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_1"/><span class="lab"><font color="Green" title="E50">A94</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_1"/><span class="lab"><font color="Green" title="E51">A95</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="txt" href="#E4:16_3_3"><span class="lab"><font color="Green" title="E48">A92</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:16_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A96</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <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="E4:16_3_3_1_1_1"/>
<font color="Maroon" title="c11">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> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E50">A94</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:16_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A97</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E3:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E52">A96</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E6:16_3_3_1_1_1"/><span class="lab"><font color="Green" title="E54">A98</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E50">A94</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:16_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A99</font></span>: 
<font color="Maroon" title="c11">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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E2:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E51">A95</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A100</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>) = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c11">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E5:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E53">A97</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">m</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E55">A99</font></span></a>, <a class="txt" href="#E5:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E53">A97</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_3_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_1_1_1_2_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="E1:16_3_3_1_1_1_2_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_1_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A101</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</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="E3:16_3_3_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">n</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> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="txt" href="#E6:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E54">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:16_3_3_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A102</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E4:16_3_3"><span class="lab"><font color="Green" title="E48">A92</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:16_3_3_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3_3"><span class="lab"><font color="Green" title="E47">A91</font></span></a>, <a class="txt" href="#E6:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E54">A98</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:16_3_3_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A103</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E2:16_3_3_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E57">A101</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E4:16_3_3_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3_3_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E59">A103</font></span></a>, <a class="txt" href="#E4:16_3_3_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E58">A102</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/><a NAME="E8:16_3_3_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:16"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="txt" href="#E8:16_3_3_1_1_1"><span class="lab"><font color="Green" title="E56">A100</font></span></a>, <a class="ref" href="topreal1.html#D7" title="TOPREAL1:def.7">TOPREAL1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_3_3_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_3_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_3_3_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E10:16_3_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_2"/><span class="lab"><font color="Green" title="E50">A104</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_2"/><span class="lab"><font color="Green" title="E51">A105</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_3_3"><span class="lab"><font color="Green" title="E49">A93</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_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="E1:16_3_3_1_1_2_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_2_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:16_3_3_1_1_2_1_1_1"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</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="E3:16_3_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A106</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E3:16_3_3"><span class="lab"><font color="Green" title="E47">A91</font></span></a>, <a class="txt" href="#E1:16_3_3_1_1_2"><span class="lab"><font color="Green" title="E50">A104</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E4:16_3_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E53">A107</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_3"><span class="lab"><font color="Green" title="E46">A90</font></span></a>, <a class="txt" href="#E4:16_3_3"><span class="lab"><font color="Green" title="E48">A92</font></span></a>, <a class="txt" href="#E1:16_3_3_1_1_2"><span class="lab"><font color="Green" title="E50">A104</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c10">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E3:16_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A106</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c10">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A106</font></span></a>, <a class="txt" href="#E4:16_3_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E53">A107</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/><span class="kw">hence </span> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_3_1_1_2"><span class="lab"><font color="Green" title="E51">A105</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c11">m</font>)</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E1:16_3_3_1_1_2"><span class="lab"><font color="Green" title="E50">A104</font></span></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_3_3_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:16_3_3_1_1_3"/>
<font color="Maroon" title="c11">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:16_3_3_1_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_3_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E7:16_3_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c10">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c11">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> or  not <font color="Maroon" title="c10">n</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="c3">h</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:16_3"/><span class="lab"><font color="Green" title="E46">A108</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="E7:16_3"/><span class="lab"><font color="Green" title="E47">A109</font></span>: 
<font color="Maroon" title="c10">n</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="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">p3</font> = <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">p4</font> = <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_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="E1:16_3_4_1"/>
( <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_3"><span class="lab"><font color="Green" title="E47">A109</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1"/><span class="lab"><font color="Green" title="E48">A110</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1"/><span class="lab"><font color="Green" title="E49">A111</font></span>: 
<font color="Maroon" title="c4">i</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A112</font></span>: <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_4_1_1"><span class="lab"><font color="Green" title="E49">A111</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_4_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_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="E1:16_3_4_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_1"/><span class="lab"><font color="Green" title="E51">A113</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A114</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_4_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_1_1_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="E1:16_3_4_1_1_2_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_4_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A114</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p11</font> where <font color="Olive" title="b1">p11</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1_2_1_1"><span class="lab"><font color="Green" title="E51">A113</font></span></a>, <a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>;<br/><a NAME="E3:16_3_4_1_1_2_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p11</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p11</font> &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:16_3_4_1_1_2_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="txt" href="#E3:16_3_4_1_1"><span class="lab"><font color="Green" title="E50">A112</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_4_1_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p22</font> where <font color="Olive" title="b1">p22</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p22</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p22</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p22</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1_2_1_1"><span class="lab"><font color="Green" title="E51">A113</font></span></a>, <a class="ref" href="topreal3.html#T9" title="TOPREAL3:th.9">TOPREAL3:9</a></span>;<br/><a NAME="E3:16_3_4_1_1_2_1_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p11</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p11</font> &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:16_3_4_1_1_2_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="txt" href="#E3:16_3_4_1_1"><span class="lab"><font color="Green" title="E50">A112</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_4_1_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_4_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A115</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A116</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:16"><span class="lab"><font color="Green" title="E33">A20</font></span></a>, <a class="ref" href="topreal3.html#T6" title="TOPREAL3:th.6">TOPREAL3:6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_4_1_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_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="E1:16_3_4_1_1_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_4_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A116</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_4_1_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p11</font> where <font color="Olive" title="b1">p11</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A115</font></span></a>, <a class="ref" href="topreal3.html#T10" title="TOPREAL3:th.10">TOPREAL3:10</a></span>;<br/><a NAME="E3:16_3_4_1_1_2_1_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p11</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p11</font> &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:16_3_4_1_1_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="txt" href="#E3:16_3_4_1_1"><span class="lab"><font color="Green" title="E50">A112</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_4_1_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_1_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_1_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p22</font> where <font color="Olive" title="b1">p22</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p22</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p22</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p22</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E19:16"><span class="lab"><font color="Green" title="E32">A19</font></span></a>, <a class="txt" href="#E23:16"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A115</font></span></a>, <a class="ref" href="topreal3.html#T10" title="TOPREAL3:th.10">TOPREAL3:10</a></span>;<br/><a NAME="E3:16_3_4_1_1_2_1_2_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p11</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p11</font> &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p11</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E4:16_3_4_1_1_2_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_1"><span class="lab"><font color="Green" title="E48">A110</font></span></a>, <a class="txt" href="#E3:16_3_4_1_1"><span class="lab"><font color="Green" title="E50">A112</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E4:16_3_4_1_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E5:16_3_4_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_3_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_3_4_1_2"/><span class="lab"><font color="Green" title="E48">A117</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><div class="add"><a NAME="E2:16_3_4_1_2"/><span class="lab"><font color="Green" title="E49">A118</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#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:16_3_4_1_2"/>
<font color="Maroon" title="c10">n</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> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_2"><span class="lab"><font color="Green" title="E48">A117</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:16_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A119</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E2:16_3_4_1_2"><span class="lab"><font color="Green" title="E49">A118</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E5:16_3_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E6:16_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A120</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_3_4_1_2"><span class="lab"><font color="Green" title="E50">A119</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><a NAME="E7:16_3_4_1_2"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_3_4_1_2"><span class="lab"><font color="Green" title="E48">A117</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E8:16_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A121</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E6:16_3"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E9:16_3_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E10:16_3_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A122</font></span>: 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_3_4_1_2"><span class="lab"><font color="Green" title="E52">A121</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><a NAME="E11:16_3_4_1_2"/>
<font color="Maroon" title="c10">n</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E7:16_3"><span class="lab"><font color="Green" title="E47">A109</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:16_3_4_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E6:16_3"><span class="lab"><font color="Green" title="E46">A108</font></span></a>, <a class="txt" href="#E10:16_3_4_1_2"><span class="lab"><font color="Green" title="E53">A122</font></span></a>, <a class="txt" href="#E6:16_3_4_1_2"><span class="lab"><font color="Green" title="E51">A120</font></span></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E9:16_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<a NAME="E35:16"/><span class="lab"><font color="Green" title="E46">A123</font></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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:16"><span class="lab"><font color="Green" title="E46">A123</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E37:16"/><span class="lab"><font color="Green" title="E47">A124</font></span>: 
( <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">h</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c10">Mf</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Mv</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : ( 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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Mh</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</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="c3">h</font> ) </span> } </span> ;<br/>
<a NAME="E38:16"/><span class="lab"><font color="Green" title="E48">A125</font></span>: 
 <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="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</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/>
<span class="kw">thus </span><a NAME="E39:16"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="topreal4.html#R1" title="TOPREAL4:pred.1">is_S-P_arc_joining</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1,<font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:16"><span class="lab"><font color="Green" title="E45">A32</font></span></a>, <a class="txt" href="#E37:16"><span class="lab"><font color="Green" title="E47">A124</font></span></a>, <a class="ref" href="topreal4.html#D1" target="_self" title="TOPREAL4:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E49">A126</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> implies ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:16_5"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_5"/><span class="lab"><font color="Green" title="E50">A127</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X</font>
 <span class="kw">and </span><br/><a NAME="E4:16_5"/><span class="lab"><font color="Green" title="E51">A128</font></span>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</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="c3">h</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">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> <span class="kw"> such that </span><br/><a NAME="E6:16_5"/><span class="lab"><font color="Green" title="E52">A129</font></span>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c15">k</font>)
 <span class="kw">and </span><br/><a NAME="E7:16_5"/><span class="lab"><font color="Green" title="E53">A130</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E8:16_5"/><span class="lab"><font color="Green" title="E54">A131</font></span>: 
<font color="Maroon" title="c15">k</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="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_5"><span class="lab"><font color="Green" title="E51">A128</font></span></a></span>;<br/><a NAME="E9:16_5"/><span class="lab"><font color="Green" title="E55">A132</font></span>: 
<font color="Maroon" title="c15">k</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E8:16_5"><span class="lab"><font color="Green" title="E54">A131</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_5_1_1"/>
( <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_5"><span class="lab"><font color="Green" title="E54">A131</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_5_1_1_1"/><span class="lab"><font color="Green" title="E56">A133</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:16_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A134</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c15">k</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E7:16_5"><span class="lab"><font color="Green" title="E53">A130</font></span></a></span>;<br/><a NAME="E3:16_5_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">i</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="txt" href="#E22:16"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_5_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E3:16_5"><span class="lab"><font color="Green" title="E50">A127</font></span></a>, <a class="txt" href="#E6:16_5"><span class="lab"><font color="Green" title="E52">A129</font></span></a>, <a class="txt" href="#E1:16_5_1_1_1"><span class="lab"><font color="Green" title="E56">A133</font></span></a>, <a class="txt" href="#E2:16_5_1_1_1"><span class="lab"><font color="Green" title="E57">A134</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span></span><br/><span class="kw">thus </span><a NAME="E5:16_5_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E3:16_5"><span class="lab"><font color="Green" title="E50">A127</font></span></a>, <a class="txt" href="#E6:16_5"><span class="lab"><font color="Green" title="E52">A129</font></span></a>, <a class="txt" href="#E1:16_5_1_1_1"><span class="lab"><font color="Green" title="E56">A133</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_5_1_1_2"/><span class="lab"><font color="Green" title="E56">A135</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/><div class="add"><a NAME="E2:16_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A136</font></span>: 
<font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">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="E3:16_5_1_1_2"/><span class="lab"><font color="Green" title="E58">A137</font></span>: 
<font color="Maroon" title="c15">k</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="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E1:16_5_1_1_2"><span class="lab"><font color="Green" title="E56">A135</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:16_5_1_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E5:16_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_5_1_1_2"><span class="lab"><font color="Green" title="E58">A137</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:16_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A138</font></span>: 
<font color="Maroon" title="c15">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E7:16_5"><span class="lab"><font color="Green" title="E53">A130</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:16_5_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</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="#E7:16_5"><span class="lab"><font color="Green" title="E53">A130</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:16_5_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A139</font></span>: 
<font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:16"><span class="lab"><font color="Green" title="E37">A24</font></span></a>, <a class="txt" href="#E3:16_5_1_1_2"><span class="lab"><font color="Green" title="E58">A137</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A140</font></span>: <font color="Maroon" title="c14">X</font> = 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c15">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E6:16_5"><span class="lab"><font color="Green" title="E52">A129</font></span></a>, <a class="txt" href="#E6:16_5_1_1_2"><span class="lab"><font color="Green" title="E59">A138</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>
<br/>.= 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c15">k</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_5_1_1_2"><span class="lab"><font color="Green" title="E60">A139</font></span></a>, <a class="txt" href="#E6:16_5_1_1_2"><span class="lab"><font color="Green" title="E59">A138</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>
;<br/><a NAME="E10:16_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_5"><span class="lab"><font color="Green" title="E53">A130</font></span></a>, <a class="txt" href="#E9:16_5"><span class="lab"><font color="Green" title="E55">A132</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:16_5_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_5"><span class="lab"><font color="Green" title="E50">A127</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span></span><br/><a NAME="E12:16_5_1_1_2"/>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c15">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_5_1_1_2"><span class="lab"><font color="Green" title="E60">A139</font></span></a>, <a class="txt" href="#E6:16_5_1_1_2"><span class="lab"><font color="Green" title="E59">A138</font></span></a>, <a class="txt" href="#E9:16_5_1_1_2"><span class="lab"><font color="Green" title="E61">A140</font></span></a>, <a class="ref" href="topreal3.html#T17" title="TOPREAL3:th.17">TOPREAL3:17</a></span>;<br/><a NAME="E13:16_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : ( 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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_5"><span class="lab"><font color="Green" title="E53">A130</font></span></a>, <a class="txt" href="#E2:16_5_1_1_2"><span class="lab"><font color="Green" title="E57">A136</font></span></a></span>;<br/><a NAME="E14:16_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : ( 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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_5"><span class="lab"><font color="Green" title="E50">A127</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E15:16_5_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E11:16_5"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E41:16"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">hence </span><a NAME="E2:16_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16"><span class="lab"><font color="Green" title="E49">A126</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<a NAME="E42:16"/><span class="lab"><font color="Green" title="E50">A141</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">thus </span><a NAME="E43:16"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_7"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E2:16_7"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:16"><span class="lab"><font color="Green" title="E49">A126</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E44:16"/><span class="lab"><font color="Green" title="E51">A142</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_8"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_8_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="E1:16_8_1"/>
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:16"><span class="lab"><font color="Green" title="E51">A142</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_8_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:16_8_1_1"/><span class="lab"><font color="Green" title="E52">A143</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X</font>
 <span class="kw">and </span><br/><a NAME="E4:16_8_1_1"/><span class="lab"><font color="Green" title="E53">A144</font></span>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">n</font>)</span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : ( 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="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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">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> <span class="kw"> such that </span><br/><a NAME="E6:16_8_1_1"/><span class="lab"><font color="Green" title="E54">A145</font></span>: 
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c15">k</font>)
 <span class="kw">and </span><br/><a NAME="E7:16_8_1_1"/><span class="lab"><font color="Green" title="E55">A146</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E8:16_8_1_1"/><span class="lab"><font color="Green" title="E56">A147</font></span>: 
<font color="Maroon" title="c15">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_8_1_1"><span class="lab"><font color="Green" title="E53">A144</font></span></a></span>;<br/><a NAME="E9:16_8_1_1"/><span class="lab"><font color="Green" title="E57">A148</font></span>: 
<font color="Maroon" title="c15">k</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="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:16"><span class="lab"><font color="Green" title="E38">A25</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a>, <a class="txt" href="#E42:16"><span class="lab"><font color="Green" title="E50">A141</font></span></a>, <a class="txt" href="#E8:16_8_1_1"><span class="lab"><font color="Green" title="E56">A147</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:16_8_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E11:16_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">k</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"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_8_1_1"><span class="lab"><font color="Green" title="E56">A147</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:16_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A149</font></span>: 
<font color="Maroon" title="c15">k</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> <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="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_8_1_1"><span class="lab"><font color="Green" title="E55">A146</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E13:16_8_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E14:16_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c4">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16_8_1_1"><span class="lab"><font color="Green" title="E56">A147</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E15:16_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c15">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E38:16"><span class="lab"><font color="Green" title="E48">A125</font></span></a>, <a class="txt" href="#E6:16_8_1_1"><span class="lab"><font color="Green" title="E54">A145</font></span></a>, <a class="txt" href="#E12:16_8_1_1"><span class="lab"><font color="Green" title="E58">A149</font></span></a>, <a class="ref" href="topreal3.html#T18" title="TOPREAL3:th.18">TOPREAL3:18</a></span>;<br/><a NAME="E16:16_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</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="c3">h</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_8_1_1"><span class="lab"><font color="Green" title="E55">A146</font></span></a>, <a class="txt" href="#E9:16_8_1_1"><span class="lab"><font color="Green" title="E57">A148</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E17:16_8_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_8_1_1"><span class="lab"><font color="Green" title="E52">A143</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="16_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_8_1_2"/><span class="lab"><font color="Green" title="E52">A150</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c1">p</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font></span><br/><div class="add"><a NAME="E2:16_8_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c4">i</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c3">h</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</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="c3">h</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E28:16"><span class="lab"><font color="Green" title="E39">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_8_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:16"><span class="lab"><font color="Green" title="E43">A30</font></span></a>, <a class="txt" href="#E1:16_8_1_2"><span class="lab"><font color="Green" title="E52">A150</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E46:16"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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