<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a>  &amp; <font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:27"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:27"/><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green">E39</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:27"/><i><font color="Green">E46</font></i>: 
(  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">h1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E11:27"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">h1</font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">Q</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="uniform1.html#T15">UNIFORM1:15</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">h1</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E49</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E15:27"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g2</font> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>and </b><br/><a NAME="E16:27"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E17:27"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<a NAME="E18:27"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">h2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E20:27"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; 5 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <font color="Maroon">h2</font> is <a href="goboard1.html#V1">increasing</a> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">Q</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="ref" href="uniform1.html#T15">UNIFORM1:15</a></i>;<br/>
<a NAME="E21:27"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:27"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">h2</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E23:27"/>
<font color="Maroon">h1</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h11</font> = <font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E25:27"/><i><font color="Green">E96</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:27"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h11</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E27:27"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E28:27"/>
<font color="Maroon">h2</font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h21</font> = <font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h2</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_2.html#T36">FINSEQ_2:36</a></i>;<br/>
<a NAME="E30:27"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E31:27"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E32:27"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E33:27"/><i><font color="Green">E102</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:27"/><b>then </b><i><font color="Green">E103</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E35:27"/><i><font color="Green">E104</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E36:27"/><i><font color="Green">E105</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E37:27"/><b>then </b><i><font color="Green">E106</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E38:27"/><i><font color="Green">E107</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E15:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E39:27"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E40:27"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E41:27"/><i><font color="Green">E110</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E42:27"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E43:27"/><b>then </b><i><font color="Green">E112</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E44:27"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E45:27"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan7.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E46:27"/><b>then </b><i><font color="Green">E114</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E13:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<a NAME="E47:27"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h0</font> = <font color="Maroon">h11</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E49:27"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="jordan7.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon">h0</font>
;<br/>

<b>thus </b><a NAME="E50:27"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>

<a NAME="E51:27"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E52:27"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E53:27"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E54:27"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<a NAME="E55:27"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E56:27"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E57:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E58:27"/><b>then </b><i><font color="Green">E122</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E59:27"/>
3 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E60:27"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E61:27"/><b>then </b><i><font color="Green">E123</font></i>: 
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E62:27"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 2
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E63:27"/><i><font color="Green">E125</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E64:27"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E65:27"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E66:27"/><b>then </b><i><font color="Green">E128</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E67:27"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E68:27"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E69:27"/><b>then </b><i><font color="Green">E131</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E70:27"/><b>then </b><i><font color="Green">E132</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E71:27"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E72:27"/><i><font color="Green">E134</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E73:27"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E74:27"/><b>then </b><i><font color="Green">E136</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E75:27"/><i><font color="Green">E137</font></i>: 
2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E76:27"/><b>then </b><i><font color="Green">E138</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E77:27"/><i><font color="Green">E139</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E78:27"/><b>then </b><i><font color="Green">E140</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E79:27"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E80:27"/><i><font color="Green">E142</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E81:27"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E82:27"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E83:27"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">h11</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>
<a NAME="E84:27"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">h21</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>
<b>thus </b><a NAME="E85:27"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">h0</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font> &amp; <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">nx</font> = <font color="Maroon">x</font>, <font color="Maroon">ny</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:27_1"/><i><font color="Green">E149</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> &amp; <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:27_1"/><i><font color="Green">E150</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> &amp; <font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1"/>
( <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">nx</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1"/>
<font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_1"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><i><font color="Green">E153</font></i>: <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> = 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">nx</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:27_1_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan7.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_2"/>
( <font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">ny</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_1"/>
<font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2_1"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_1_2_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> = 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font>
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ny</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E5:27_1_1_1_2_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ny</font>
 <b>by </b><i>, <a class="ref" href="jordan7.html#D1" target="_self">Def1</a>, , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:27_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">ny</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_1_1_1_2_2"/><i><font color="Green">E159</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> &amp; <font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_1_1_1_2_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_1_1_1_2_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_1_2_2"/><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E164</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:27_1_1_1_2_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_1_1_1_2_2"/><i><font color="Green">E166</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_1_1_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E167</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_1_1_1_2_2"/>
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E15:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_1_1_1_2_2"/><i><font color="Green">E170</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E17:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/><a NAME="E20:27_1_1_1_2_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E21:27_1_1_1_2_2"/><b>then </b><i><font color="Green">E172</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/><a NAME="E22:27_1_1_1_2_2"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E23:27_1_1_1_2_2"/><i><font color="Green">E174</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_2_2_1"/>
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2_1_1"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E3:27_1_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E7:27_1_1_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_2_2_1_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_1_2_2_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , , <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">nx</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_1_1_2"/><i><font color="Green">E177</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> &amp; <font color="Maroon">nx</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_1_1_2"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">nx</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_1_1_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">nx</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_1_1_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">nx</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_2"/><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_1_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:27_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_1_1_2"/><i><font color="Green">E184</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:27_1_1_2"/><b>then </b><i><font color="Green">E185</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_1_1_2"/>
<font color="Maroon">nx</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_1_1_2"/><b>then </b><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E15:27_1_1_2"/><b>then </b><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_1_1_2"/><i><font color="Green">E188</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E17:27_1_1_2"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_1_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_1_1_2"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nx</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T25">FUNCT_1:25</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_2_1"/>
( <font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">ny</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1"/>
<font color="Maroon">ny</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2_1_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_1_1_2_1_1"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> = 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font>
<b>by </b><i>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ny</font></span>)</span>
<b>by </b><i>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:27_1_1_2_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="jordan7.html#D1" target="_self">Def1</a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_1_1_2_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:27_1_1_2_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/><a NAME="E8:27_1_1_2_1_1"/><i><font color="Green">E194</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E9:27_1_1_2_1_1"/><i><font color="Green">E195</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_2_1_1_2"/>
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1_2_1"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E6:27_1_1_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_1_2_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E3:27_1_1_2_1_1_2_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:27_1_1_2_1_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_2_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">ny</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_1_1_2_1_2"/><b>then </b><i><font color="Green">E197</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_1_1_2_1_2"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_1_1_2_1_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:27_1_1_2_1_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_1_1_2_1_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ny</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_1_1_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E9:27_1_1_2_1_2"/><b>then </b><i><font color="Green">E202</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_1_1_2_1_2"/>
<font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E11:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:27_1_1_2_1_2"/><b>then </b><i><font color="Green">E203</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E13:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E15:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E16:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E17:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E18:27_1_1_2_1_2"/><b>then </b>
<font color="Maroon">nx</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ny</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 ;<br/><a NAME="E19:27_1_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">nx</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">ny</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="xcmplx_1.html#T29">XCMPLX_1:29</a></i>;<br/><a NAME="E20:27_1_1_2_1_2"/><b>then </b>
<font color="Maroon">nx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ny</font>
 ;<br/><b>hence </b><a NAME="E21:27_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E86:27"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E11:27"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, , , <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E87:27"/>
<span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E88:27"/><b>then </b><i><font color="Green">E205</font></i>: 
5 <a href="nat_1.html#K1">+</a> 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E89:27"/>
8 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>

<a NAME="E90:27"/><i><font color="Green">E206</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E91:27"/><i><font color="Green">E207</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T28">JORDAN3:28</a></i>;<br/>
<a NAME="E92:27"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E93:27"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E94:27"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">h11</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E22:27"><i><font color="Green">E94</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E95:27"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h0</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<b>hence </b><a NAME="E96:27"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>

<b>thus </b><a NAME="E97:27"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> holds <br/> <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E208</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 ;<br/>

<a NAME="E2:27_2"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:27_2"/><i><font color="Green">E210</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:27_2"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1_1"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_2_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_2_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_2_1_1"/><b>then </b><i><font color="Green">E216</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_2_1_1"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_2_1_1"/><b>then </b><i><font color="Green">E218</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_2_1_1"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E9:27_2_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:27_2_1_1"/><b>then </b><i><font color="Green">E221</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E11:27_2_1_1"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:27_2_1_1"/><b>then </b><i><font color="Green">E223</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E13:27_2_1_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E14:27_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:27_2_1_1"/><b>then </b><i><font color="Green">E225</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:27_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:27_2_1_1"/><b>then </b><i><font color="Green">E226</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E18:27_2_1_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E19:27_2_1_1"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_2_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, </i>;<br/><b>hence </b><a NAME="E21:27_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_2_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_2_1_2_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_2_1_2_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_2_1_2_1_1"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_2_1_2_1_1"/><i><font color="Green">E236</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E238</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:27_2_1_2_1_1"/><i><font color="Green">E239</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_2_1_2_1_1"/><i><font color="Green">E240</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_2_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E242</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E18:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E243</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E19:27_2_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E20:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E244</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E246</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><i><font color="Green">E247</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/><i><font color="Green">E248</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E26:27_2_1_2_1_1"/><i><font color="Green">E249</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E250</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E28:27_2_1_2_1_1"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E252</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E30:27_2_1_2_1_1"/><i><font color="Green">E253</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E31:27_2_1_2_1_1"/><i><font color="Green">E254</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E32:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E255</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E33:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E34:27_2_1_2_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E35:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E36:27_2_1_2_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E37:27_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E38:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E39:27_2_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E40:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E261</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E41:27_2_1_2_1_1"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E42:27_2_1_2_1_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E43:27_2_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E44:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E45:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E46:27_2_1_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E47:27_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E48:27_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E49:27_2_1_2_1_1"/><b>then </b><i><font color="Green">E266</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><i><font color="Green">E267</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1_2_1_1_3"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_2_1_2_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_2_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:27_2_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E5:27_2_1_2_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E51:27_2_1_2_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, </i>;<br/><b>hence </b><a NAME="E52:27_2_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_2_1_2_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E269</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E3:27_2_1_2_1_2"/><i><font color="Green">E270</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_2_1_2_1_2"/><i><font color="Green">E271</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_2_1_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a></i>;<br/><a NAME="E6:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E7:27_2_1_2_1_2"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E8:27_2_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E10:27_2_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:27_2_1_2_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
0

;<br/><a NAME="E14:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E276</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E15:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E277</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><i><font color="Green">E278</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>
;<br/><a NAME="E17:27_2_1_2_1_2"/><i><font color="Green">E279</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E280</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E19:27_2_1_2_1_2"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_2_1_2_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E282</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:27_2_1_2_1_2"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E23:27_2_1_2_1_2"/>
2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:27_2_1_2_1_2"/><b>then </b><i><font color="Green">E284</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:27_2_1_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a></i>;<br/><a NAME="E26:27_2_1_2_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E27:27_2_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E98:27"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:27_3"/><i><font color="Green">E285</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:27_3"/><b>then </b><i><font color="Green">E286</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:27_3"/><i><font color="Green">E287</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:27_3"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_1"/><b>then </b><i><font color="Green">E290</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_3_1_1"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_3_1_1"/><i><font color="Green">E292</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_3_1_1"/><b>then </b><i><font color="Green">E293</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_3_1_1"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_3_1_1"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1"/><b>then </b><i><font color="Green">E296</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E10:27_3_1_1"/><i><font color="Green">E298</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:27_3_1_1"/><b>then </b><i><font color="Green">E299</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E12:27_3_1_1"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:27_3_1_1"/><b>then </b><i><font color="Green">E301</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_3_1_1"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E15:27_3_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_3_1_1"/><b>then </b><i><font color="Green">E303</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E17:27_3_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_3_1_1"/><b>then </b><i><font color="Green">E304</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E19:27_3_1_1"/><i><font color="Green">E305</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_3_1_1"/><i><font color="Green">E306</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E21:27_3_1_1"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E22:27_3_1_1"/><i><font color="Green">E308</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E24:27_3_1_1"/><i><font color="Green">E310</font></i>: 
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_1_1"/><i><font color="Green">E311</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:27_3_1_1_1"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_1_1"/><i><font color="Green">E313</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_1_1"/><i><font color="Green">E314</font></i>: 
( ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_1_1"/><i><font color="Green">E315</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E316</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E13:27"><i><font color="Green">E49</font></i></a>, , , <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><i><font color="Green">E317</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_1"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E4:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E318</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><a NAME="E5:27_3_1_1_1_1_1_1"/><i><font color="Green">E319</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><a NAME="E6:27_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E320</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_1_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E321</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green">E322</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E13:27"><i><font color="Green">E49</font></i></a>, , , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><a NAME="E5:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E323</font></i>: 
( ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><a NAME="E6:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="jordan6.html#T69">JORDAN6:69</a></i>;<br/><div><i><font color="Green">E325</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green">E327</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E12:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green">E330</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E13:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E331</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E15:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a></i>;<br/><a NAME="E16:27_3_1_1_1_1_1_1_2_1"/><i><font color="Green">E332</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E17:27_3_1_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E333</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><a NAME="E18:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>hence </b><a NAME="E19:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>thus </b><a NAME="E20:27_3_1_1_1_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2"/><i><font color="Green">E334</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green">E335</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_1_2_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green">E336</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a></i>;<br/><b>take </b><font color="Maroon">rz</font> = <font color="Maroon">rz</font>;<br/><a NAME="E6:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1_1_1_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green">E338</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green">E339</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:27_3_1_1_1_1_1_1_2_2_1_1_1"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
<font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>hence </b><a NAME="E12:27_3_1_1_1_1_1_1_2_2_1_1_1"/>
( <font color="Maroon">rz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">rz</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rz</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_1_2_2_1_1_2"/><i><font color="Green">E341</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:27_3_1_1_1_1_1_1_2_2_1_1_2"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>thus </b><a NAME="E3:27_3_1_1_1_1_1_1_2_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_1_2_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E4:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2"/><i><font color="Green">E342</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E343</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E13:27"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2"/><i><font color="Green">E344</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><div><i><font color="Green">E345</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_2_1"/><i><font color="Green">E346</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E3"><i><font color="Green">Lemma31</font></i></a>, <a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E5:27_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E347</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_2_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1"/><i><font color="Green">E348</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_1"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green">E349</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><div><i><font color="Green">E350</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green">E351</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a></i>;<br/><b>take </b><font color="Maroon">rz</font> = <font color="Maroon">rz</font>;<br/><a NAME="E6:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_3_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E353</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_3_1_1_1_1_1_2_2_1_1_1_1"/><i><font color="Green">E354</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:27_3_1_1_1_1_1_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E355</font></i>: 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><a NAME="E11:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
<font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>hence </b><a NAME="E12:27_3_1_1_1_1_1_2_2_1_1_1_1"/>
( <font color="Maroon">rz</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">rz</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rz</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_1_1_1_2"><b>case </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_1_1_1_2"/><i><font color="Green">E356</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="jordan6.html#T69">JORDAN6:69</a></i>;<br/><a NAME="E3:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:27_3_1_1_1_1_1_2_2_1_1_1_2"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_1_1_1_1_2_2_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E4:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_2_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_1_1_1_1_2_2_2"><b>suppose </b></a><a NAME="E1:27_3_1_1_1_1_1_2_2_2"/><i><font color="Green">E357</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_1_1_1_1_2_2_2"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_1_1_1_1_2_2_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan7.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:27_3_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:27_3_1_1"/>
<font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_1_2"/><i><font color="Green">E358</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_1_2"/><i><font color="Green">E359</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_1_2"/><i><font color="Green">E361</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E8:27_3_1_1_2"/><i><font color="Green">E362</font></i>: 
( <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_1_2"/><i><font color="Green">E363</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E10:27_3_1_1_2"/><i><font color="Green">E364</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:27_3_1_1_2"/><b>then </b><i><font color="Green">E365</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, </i>;<br/>
<a NAME="E12:27_3_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, </i>;<br/>
<a NAME="E13:27_3_1_1_2"/><b>then </b><i><font color="Green">E366</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E14:27_3_1_1_2"/><i><font color="Green">E367</font></i>: 
not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E15:27_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:27_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E26:27_3_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E27:27_3_1_1"/>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_2"><b>suppose </b></a><a NAME="E1:27_3_1_2"/><i><font color="Green">E368</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_2"/><b>then </b><i><font color="Green">E369</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_3_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_3_1_2"/><b>then </b><i><font color="Green">E370</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_3_1_2"/><i><font color="Green">E371</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_3_1_2"/><i><font color="Green">E372</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_3_1_2"/><b>then </b><i><font color="Green">E373</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_3_1_2"/><i><font color="Green">E374</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:27_3_1_2"/><b>then </b><i><font color="Green">E375</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_3_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:27_3_1_2"/><b>then </b><i><font color="Green">E376</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:27_3_1_2"/><i><font color="Green">E377</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_3_1_2"/><i><font color="Green">E378</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_3_1_2"/><b>then </b><i><font color="Green">E379</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_3_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:27_3_1_2"/><b>then </b><i><font color="Green">E380</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E18:27_3_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E19:27_3_1_2"/><b>then </b><i><font color="Green">E381</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:27_3_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E21:27_3_1_2"/><b>then </b><i><font color="Green">E382</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E22:27_3_1_2"/><b>then </b><i><font color="Green">E383</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_3_1_2"/><i><font color="Green">E384</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E24:27_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E26:27_3_1_2"/><b>then </b><i><font color="Green">E385</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E27:27_3_1_2"/><i><font color="Green">E386</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E28:27_3_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E29:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E30:27_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E31:27_3_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E32:27_3_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_3_1_2"/><b>then </b><i><font color="Green">E387</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E34:27_3_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E35:27_3_1_2"/><b>then </b><i><font color="Green">E388</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E36:27_3_1_2"/><i><font color="Green">E389</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><i><font color="Green">E390</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E38:27_3_1_2"/><i><font color="Green">E391</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E39:27_3_1_2"/><i><font color="Green">E392</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E40:27_3_1_2"/><b>then </b><i><font color="Green">E393</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E41:27_3_1_2"/><i><font color="Green">E394</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E42:27_3_1_2"/><i><font color="Green">E395</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E43:27_3_1_2"/><b>then </b><i><font color="Green">E396</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E44:27_3_1_2"/><b>then </b><i><font color="Green">E397</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E45:27_3_1_2"/><i><font color="Green">E398</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E46:27_3_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E47:27_3_1_2"/><b>then </b><i><font color="Green">E399</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E48:27_3_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E49:27_3_1_2"/><b>then </b><i><font color="Green">E400</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E50:27_3_1_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E51:27_3_1_2"/><b>then </b><i><font color="Green">E401</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E52:27_3_1_2"/><i><font color="Green">E402</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E53:27_3_1_2"/><i><font color="Green">E403</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E54:27_3_1_2"/><i><font color="Green">E404</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E55:27_3_1_2"/><i><font color="Green">E405</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E56:27_3_1_2"/><i><font color="Green">E406</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E57:27_3_1_2"/><i><font color="Green">E407</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E58:27_3_1_2"/><i><font color="Green">E408</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E60:27_3_1_2"/><i><font color="Green">E409</font></i>: 
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_2_2"/><i><font color="Green">E410</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:27_3_1_2_2"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_2_2"/><i><font color="Green">E411</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_2_2"/><i><font color="Green">E412</font></i>: 
( ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_2_2"/><i><font color="Green">E413</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:27_3_1_2_2"/><i><font color="Green">E414</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E9:27_3_1_2_2"/><i><font color="Green">E415</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_1"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_2_2_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_2_2"/><b>then </b><i><font color="Green">E416</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E12:27_3_1_2_2"/><i><font color="Green">E417</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E13:27_3_1_2_2"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green">E419</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_2"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_2_2_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:27_3_1_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:27_3_1_2_2"/><b>then </b><i><font color="Green">E420</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:27_3_1_2_2"/><i><font color="Green">E421</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green">E422</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_2_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_2_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_2_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_2_2_3"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_2_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , , <a class="ref" href="jordan6.html#T69">JORDAN6:69</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:27_3_1_2_2"/><b>then </b><i><font color="Green">E423</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:27_3_1_2_2"/><i><font color="Green">E424</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E22:27_3_1_2_2"/><i><font color="Green">E425</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E23:27_3_1_2_2"/><i><font color="Green">E426</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E24:27_3_1_2_2"/>
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:27_3_1_2_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:27_3_1_2_2"/><b>then </b>
( <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> )
 
<b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, , , , , , <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/>
<a NAME="E27:27_3_1_2_2"/><b>then </b>
<font color="Maroon">rz</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E28:27_3_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E61:27_3_1_2"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_2_3"/><i><font color="Green">E427</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_2_3"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_2_3"/><i><font color="Green">E429</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_3_1_2_3"/><i><font color="Green">E430</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_2_3"/><i><font color="Green">E431</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green">E432</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_2_3_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_2_3_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">sy</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E21:27"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_2_3"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E12:27_3_1_2_3"/><i><font color="Green">E434</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, </i>;<br/>
<a NAME="E13:27_3_1_2_3"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, </i>;<br/>
<a NAME="E14:27_3_1_2_3"/><b>then </b>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, , , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E15:27_3_1_2_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E16:27_3_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E62:27_3_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E63:27_3_1_2"/>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_3_1_3"><b>suppose </b></a><a NAME="E1:27_3_1_3"/><i><font color="Green">E435</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_3_1_3"/><b>then </b><i><font color="Green">E436</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_3_1_3"/><i><font color="Green">E437</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_3_1_3"/><i><font color="Green">E438</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_3_1_3"/><i><font color="Green">E439</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_3_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_3_1_3"/><b>then </b><i><font color="Green">E440</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E8:27_3_1_3"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:27_3_1_3"/><b>then </b><i><font color="Green">E441</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E10:27_3_1_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E11:27_3_1_3"/><b>then </b><i><font color="Green">E442</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:27_3_1_3"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_3_1_3"/><b>then </b><i><font color="Green">E443</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_3_1_3"/><b>then </b><i><font color="Green">E444</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_3_1_3"/><i><font color="Green">E445</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E16:27_3_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:27_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_3_1_3"/><b>then </b><i><font color="Green">E446</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E19:27_3_1_3"/><i><font color="Green">E447</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E20:27_3_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E21:27_3_1_3"/><b>then </b><i><font color="Green">E448</font></i>: 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E22:27_3_1_3"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_3_1_3"/><b>then </b><i><font color="Green">E449</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E450</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E25:27_3_1_3"/><b>then </b><i><font color="Green">E451</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><i><font color="Green">E452</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E27:27_3_1_3"/><i><font color="Green">E453</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:27_3_1_3"/><b>then </b><i><font color="Green">E454</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E29:27_3_1_3"/><i><font color="Green">E455</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E30:27_3_1_3"/><i><font color="Green">E456</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_3_1_3"/><i><font color="Green">E457</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E32:27_3_1_3"/><i><font color="Green">E458</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E33:27_3_1_3"/><i><font color="Green">E459</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:27_3_1_3"/><b>then </b><i><font color="Green">E460</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E35:27_3_1_3"/><i><font color="Green">E461</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E36:27_3_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:27_3_1_3"/><b>then </b><i><font color="Green">E462</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E38:27_3_1_3"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E39:27_3_1_3"/><b>then </b><i><font color="Green">E463</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E40:27_3_1_3"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:27_3_1_3"/><b>then </b><i><font color="Green">E464</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E42:27_3_1_3"/><i><font color="Green">E465</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E43:27_3_1_3"/><i><font color="Green">E466</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E44:27_3_1_3"/><i><font color="Green">E467</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E45:27_3_1_3"/><i><font color="Green">E468</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E46:27_3_1_3"/><i><font color="Green">E469</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E47:27_3_1_3"/><i><font color="Green">E470</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><b>then reconsider </b><font color="Maroon">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E49:27_3_1_3"/><i><font color="Green">E471</font></i>: 
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_3_3"/><i><font color="Green">E472</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:27_3_1_3_3"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , , <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E3:27_3_1_3_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_3_1_3_3"/><i><font color="Green">E473</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> )
 ;<br/>
<a NAME="E6:27_3_1_3_3"/><i><font color="Green">E474</font></i>: 
( ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E7:27_3_1_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E8:27_3_1_3_3"/><b>then </b><i><font color="Green">E475</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:27_3_1_3_3"/><i><font color="Green">E476</font></i>: 
not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E10:27_3_1_3_3"/><i><font color="Green">E477</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_1"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_3_3_1"/><b>then </b>
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_3_3_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , , <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:27_3_1_3_3"/><b>then </b><i><font color="Green">E478</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 
<b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E13:27_3_1_3_3"/><i><font color="Green">E479</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E14:27_3_1_3_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E15:27_3_1_3_3"/><b>then </b><i><font color="Green">E480</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T10" target="_self">Th10</a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><i><font color="Green">E481</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_2"/><i><font color="Green">E482</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_3_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_3_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E483</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_3_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_3_3_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_3_1_3_3_3"/><b>then </b><i><font color="Green">E484</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_3_1_3_3_3"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="jordan6.html#T69">JORDAN6:69</a></i>;<br/><b>hence </b><a NAME="E5:27_3_1_3_3_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:27_3_1_3_3"/><b>then </b><i><font color="Green">E485</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E19:27_3_1_3_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:27_3_1_3_3"/><i><font color="Green">E486</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E23:27_3_1_3_3"/><i><font color="Green">E487</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E24:27_3_1_3_3"/>
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:27_3_1_3_3"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E26:27_3_1_3_3"/><b>then </b>
( <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> )
 
<b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, , , , <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E27:27_3_1_3_3"/><b>then </b>
<font color="Maroon">rz</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E28:27_3_1_3_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E50:27_3_1_3"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_3_1_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_3_1_3_4"/><i><font color="Green">E488</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_3_1_3_4"/><i><font color="Green">E489</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a></i>;<br/>
<a NAME="E7:27_3_1_3_4"/><i><font color="Green">E490</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_3_1_3_4"/><i><font color="Green">E491</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_3_1_3_4"/><i><font color="Green">E492</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, , <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<div><i><font color="Green">E493</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3_1_3_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_3_1_3_4_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_3_1_3_4_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">sy</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E21:27"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:27_3_1_3_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:27_3_1_3_4"/><i><font color="Green">E494</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , , <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E12:27_3_1_3_4"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, </i>;<br/>
<a NAME="E13:27_3_1_3_4"/><b>then </b><i><font color="Green">E495</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E14:27_3_1_3_4"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, </i>;<br/>
<a NAME="E15:27_3_1_3_4"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:27_3_1_3_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:27_3_1_3_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E51:27_3_1_3"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E52:27_3_1_3"/>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E99:27"/>
for <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>  st <font color="Olive">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font> holds <br/> <a href="tbsp_1.html#K2">diameter</a> <font color="Olive">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:27_4"/><i><font color="Green">E496</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font>
 ;<br/>

<b>set </b><font color="Maroon">i</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>;<br/>
<a NAME="E2:27_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E3:27_4"/><b>then </b><i><font color="Green">E497</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E4:27_4"/><i><font color="Green">E498</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E5:27_4"/><i><font color="Green">E499</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E6:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_4"/><b>then </b><i><font color="Green">E500</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E8:27_4"/><i><font color="Green">E501</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E9:27_4"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E10:27_4"/><b>then </b><i><font color="Green">E502</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:27_4"/><b>then </b><i><font color="Green">E503</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E12:27_4"/><i><font color="Green">E504</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E13:27_4"/><b>then </b><i><font color="Green">E505</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E14:27_4"/><i><font color="Green">E506</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E15:27_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:27_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E17:27_4"/><b>then </b><i><font color="Green">E507</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:27_4"/><i><font color="Green">E508</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E19:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:27_4"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<a NAME="E21:27_4"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E22:27_4"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E23:27_4"/><b>then </b><i><font color="Green">E509</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:27_4"/><i><font color="Green">E510</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/>
<i><font color="Green">E511</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/>
<a NAME="E26:27_4"/><i><font color="Green">E512</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:27_4"/><b>then </b><i><font color="Green">E513</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E28:27_4"/><i><font color="Green">E514</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:27_4"/><b>then </b><i><font color="Green">E515</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E30:27_4"/><i><font color="Green">E516</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E31:27_4"/><i><font color="Green">E517</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><i><font color="Green">E518</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_4_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:27_4_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_4_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E4:27_4_2"/><b>then </b>
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:27_4_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E33:27_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E34:27_4"/><b>then </b><i><font color="Green">E519</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E35:27_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E36:27_4"/><b>then </b><i><font color="Green">E520</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E37:27_4"/><i><font color="Green">E521</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E38:27_4"/><i><font color="Green">E522</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E39:27_4"/><i><font color="Green">E523</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E40:27_4"/><i><font color="Green">E524</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Q1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/>
<a NAME="E42:27_4"/><i><font color="Green">E525</font></i>: 
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_4_3"/><i><font color="Green">E526</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:27_4_3"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E3:27_4_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Olive">p</font>,<font color="Maroon">P</font> or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:27_4_3"/><i><font color="Green">E527</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">P</font> or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) )
 ;<br/>
<a NAME="E6:27_4_3"/><i><font color="Green">E528</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:27_4_3"/><i><font color="Green">E529</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:27_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:27_4_3"/><b>then </b>
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E10:27_4_3"/><b>then </b><i><font color="Green">E530</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E11:27_4_3"/><i><font color="Green">E531</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_3_1_1"/>
( (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp; ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or  not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) ) or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:27_4_3_1_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp; ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or  not <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) )
 ;<br/><div class="add"><a NAME="E2:27_4_3_1_1_1"/><b>then </b><i><font color="Green">E532</font></i>: 
( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:27_4_3_1_1_1"/><i><font color="Green">E533</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">rz</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a></i>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><a NAME="E6:27_4_3_1_1_1"/><i><font color="Green">E534</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font> &amp; <font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:27_4_3_1_1_1"/><b>then </b><i><font color="Green">E535</font></i>: 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rz</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E1"><i><font color="Green">Lemma23</font></i></a></i>;<br/><b>thus </b><a NAME="E8:27_4_3_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a></i>;<br/><a NAME="E9:27_4_3_1_1_1"/>
<font color="Maroon">rz</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><b>hence </b><a NAME="E10:27_4_3_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:27_4_3_1_1_2"/>
( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_3_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E21:27"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:27_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:27_4"/>
<font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:27_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_4_4"/><i><font color="Green">E536</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:27_4_4"/><i><font color="Green">E537</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">sy</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a></i>;<br/>
<a NAME="E7:27_4_4"/><i><font color="Green">E538</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:27_4_4"/><i><font color="Green">E539</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sy</font> &amp; <font color="Maroon">sy</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:27_4_4"/><i><font color="Green">E540</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E10:27_4_4"/><i><font color="Green">E541</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E11:27_4_4"/><i><font color="Green">E542</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p1</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_4_4_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_4_1_1_1"><b>case </b></a><a NAME="E1:27_4_4_1_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_4_1_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_4_4_1_1_2"><b>case </b></a><a NAME="E1:27_4_4_1_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:27_4_4_1_1_2"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:27_4_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Olive">p</font>,<font color="Maroon">P</font> or ( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E14:27_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:27_4"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E45:27_4"/>
 <a href="tbsp_1.html#K2">diameter</a> <font color="Maroon">W</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, </i>;<br/>


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

<a NAME="E100:27"/><i><font color="Green">E543</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> holds <br/> <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:27_5"/><i><font color="Green">E544</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 ;<br/>

<a NAME="E2:27_5"/><b>then </b><i><font color="Green">E545</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:27_5"/><i><font color="Green">E546</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:27_5"/><b>then </b><i><font color="Green">E547</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:27_5"/><i><font color="Green">E548</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_5_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_1"><b>suppose </b></a><a NAME="E1:27_5_1_1"/><i><font color="Green">E549</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_5_1_1"/><b>then </b><i><font color="Green">E550</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_5_1_1"/><i><font color="Green">E551</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_5_1_1"/><i><font color="Green">E552</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_5_1_1"/><b>then </b><i><font color="Green">E553</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_5_1_1"/><i><font color="Green">E554</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_5_1_1"/><b>then </b><i><font color="Green">E555</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_5_1_1"/><i><font color="Green">E556</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:27_5_1_1"/><b>then </b><i><font color="Green">E557</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E10:27_5_1_1"/><i><font color="Green">E558</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:27_5_1_1"/><b>then </b><i><font color="Green">E559</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E12:27_5_1_1"/><i><font color="Green">E560</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E13:27_5_1_1"/><i><font color="Green">E561</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_5_1_1"/><i><font color="Green">E562</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E15:27_5_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_5_1_1"/><b>then </b><i><font color="Green">E563</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E17:27_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_5_1_1"/><b>then </b><i><font color="Green">E564</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E19:27_5_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E20:27_5_1_1"/><b>then </b><i><font color="Green">E565</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E21:27_5_1_1"/><i><font color="Green">E566</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E22:27_5_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, </i>;<br/><b>hence </b><a NAME="E23:27_5_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2"><b>suppose </b></a><a NAME="E1:27_5_1_2"/><i><font color="Green">E567</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_5_1_2_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_5_1_2_1_1"/><i><font color="Green">E568</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E569</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_5_1_2_1_1"/><i><font color="Green">E570</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_5_1_2_1_1"/><i><font color="Green">E571</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E572</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_5_1_2_1_1"/><i><font color="Green">E573</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E7:27_5_1_2_1_1"/><i><font color="Green">E574</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E575</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_5_1_2_1_1"/><i><font color="Green">E576</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E577</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E12:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E578</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E14:27_5_1_2_1_1"/><i><font color="Green">E579</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_5_1_2_1_1"/><i><font color="Green">E580</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E581</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E17:27_5_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E582</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E19:27_5_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E20:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E583</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E584</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E22:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E585</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E24:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E586</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E25:27_5_1_2_1_1"/><i><font color="Green">E587</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E26:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E588</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E589</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/><a NAME="E29:27_5_1_2_1_1"/><i><font color="Green">E590</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E30:27_5_1_2_1_1"/><i><font color="Green">E591</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E31:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E592</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E32:27_5_1_2_1_1"/><i><font color="Green">E593</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E33:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E594</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E34:27_5_1_2_1_1"/><i><font color="Green">E595</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E35:27_5_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E36:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E596</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E597</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E38:27_5_1_2_1_1"/><i><font color="Green">E598</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E39:27_5_1_2_1_1"/><i><font color="Green">E599</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E40:27_5_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E41:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E600</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E42:27_5_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E43:27_5_1_2_1_1"/><b>then </b><i><font color="Green">E601</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E44:27_5_1_2_1_1"/><i><font color="Green">E602</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E45:27_5_1_2_1_1"/><i><font color="Green">E603</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E46:27_5_1_2_1_1"/><i><font color="Green">E604</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E47:27_5_1_2_1_1"/><i><font color="Green">E605</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, , <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E48:27_5_1_2_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, </i>;<br/><b>hence </b><a NAME="E49:27_5_1_2_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_5_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_5_1_2_1_2"/><i><font color="Green">E606</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E607</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_5_1_2_1_2"/><i><font color="Green">E608</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_5_1_2_1_2"/><i><font color="Green">E609</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_5_1_2_1_2"/><i><font color="Green">E610</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_5_1_2_1_2"/><i><font color="Green">E611</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E7:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E612</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E613</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_5_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E614</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E12:27_5_1_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E13:27_5_1_2_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E615</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E15:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E616</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E17:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E617</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E618</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E20:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E619</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><i><font color="Green">E620</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E22:27_5_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E621</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E622</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_5_1_2_1_2"/><i><font color="Green">E623</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E26:27_5_1_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E624</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E28:27_5_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E29:27_5_1_2_1_2"/><b>then </b><i><font color="Green">E625</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E30:27_5_1_2_1_2"/><i><font color="Green">E626</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_5_1_2_1_2"/><i><font color="Green">E627</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, </i>;<br/><a NAME="E32:27_5_1_2_1_2"/><i><font color="Green">E628</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, , <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>thus </b><a NAME="E33:27_5_1_2_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E101:27"/><i><font color="Green">E629</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> holds <br/>(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_6"/><i><font color="Green">E630</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:27_6"/><i><font color="Green">E631</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 ;<br/>

<a NAME="E3:27_6"/><i><font color="Green">E632</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:27_6"/><i><font color="Green">E633</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:27_6"/><i><font color="Green">E634</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:27_6"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:27_6"/><b>then </b><i><font color="Green">E635</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E8:27_6"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:27_6"/><b>then </b><i><font color="Green">E636</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_6_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_1"><b>suppose </b></a><a NAME="E1:27_6_1_1"/><i><font color="Green">E637</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_6_1_1"/><b>then </b><i><font color="Green">E638</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_6_1_1"/><i><font color="Green">E639</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E4:27_6_1_1"/><i><font color="Green">E640</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_6_1_1"/><b>then </b><i><font color="Green">E641</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_6_1_1"/><i><font color="Green">E642</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_6_1_1"/><i><font color="Green">E643</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:27_6_1_1"/><b>then </b><i><font color="Green">E644</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E9:27_6_1_1"/><i><font color="Green">E645</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:27_6_1_1"/><i><font color="Green">E646</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E11:27_6_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:27_6_1_1"/><b>then </b><i><font color="Green">E647</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E13:27_6_1_1"/><i><font color="Green">E648</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:27_6_1_1"/><i><font color="Green">E649</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E15:27_6_1_1"/><i><font color="Green">E650</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E16:27_6_1_1"/><i><font color="Green">E651</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E17:27_6_1_1"/><i><font color="Green">E652</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E18:27_6_1_1"/><i><font color="Green">E653</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:27_6_1_1"/><i><font color="Green">E654</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E20:27_6_1_1"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, </i>;<br/><b>hence </b><a NAME="E21:27_6_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2"><b>suppose </b></a><a NAME="E1:27_6_1_2"/><i><font color="Green">E655</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_6_1_2_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_6_1_2_1_1"/><i><font color="Green">E656</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E657</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_6_1_2_1_1"/><i><font color="Green">E658</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E659</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_6_1_2_1_1"/><i><font color="Green">E660</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_6_1_2_1_1"/><i><font color="Green">E661</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_6_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E662</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_6_1_2_1_1"/><i><font color="Green">E663</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E664</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, , <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E11:27_6_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E665</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E13:27_6_1_2_1_1"/><i><font color="Green">E666</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_6_1_2_1_1"/><i><font color="Green">E667</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E15:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E668</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E16:27_6_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E669</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E18:27_6_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E19:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E670</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E671</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E21:27_6_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E672</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:27_6_1_2_1_1"/><i><font color="Green">E673</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E24:27_6_1_2_1_1"/><i><font color="Green">E674</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E25:27_6_1_2_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E26:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E675</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E676</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><i><font color="Green">E677</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E29:27_6_1_2_1_1"/><i><font color="Green">E678</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E679</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E31:27_6_1_2_1_1"/><i><font color="Green">E680</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_6_1_2_1_1"/><i><font color="Green">E681</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E33:27_6_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E34:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E682</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E35:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E683</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E36:27_6_1_2_1_1"/><i><font color="Green">E684</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E37:27_6_1_2_1_1"/><i><font color="Green">E685</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E38:27_6_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E39:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E686</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E40:27_6_1_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E41:27_6_1_2_1_1"/><b>then </b><i><font color="Green">E687</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E42:27_6_1_2_1_1"/><i><font color="Green">E688</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E43:27_6_1_2_1_1"/><i><font color="Green">E689</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E44:27_6_1_2_1_1"/><i><font color="Green">E690</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E45:27_6_1_2_1_1"/><i><font color="Green">E691</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E46:27_6_1_2_1_1"/><i><font color="Green">E692</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E47:27_6_1_2_1_1"/>
( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E48:27_6_1_2_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_6_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_6_1_2_1_2"/><i><font color="Green">E693</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E694</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_6_1_2_1_2"/><i><font color="Green">E695</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_6_1_2_1_2"/><i><font color="Green">E696</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E5:27_6_1_2_1_2"/><i><font color="Green">E697</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, , <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E6:27_6_1_2_1_2"/><i><font color="Green">E698</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E7:27_6_1_2_1_2"/><i><font color="Green">E699</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_6_1_2_1_2"/><i><font color="Green">E700</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a></i>;<br/><a NAME="E9:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E701</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E10:27_6_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E702</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E12:27_6_1_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E13:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E703</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E704</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E15:27_6_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E705</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E706</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E18:27_6_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E19:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E707</font></i>: 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><a NAME="E20:27_6_1_2_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E21:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E708</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><i><font color="Green">E709</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/><a NAME="E23:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E710</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><i><font color="Green">E711</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E25:27_6_1_2_1_2"/><i><font color="Green">E712</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E26:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E713</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E27:27_6_1_2_1_2"/><i><font color="Green">E714</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E28:27_6_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E29:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E715</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E716</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E31:27_6_1_2_1_2"/><i><font color="Green">E717</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_6_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E33:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E718</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E34:27_6_1_2_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E35:27_6_1_2_1_2"/><b>then </b><i><font color="Green">E719</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E36:27_6_1_2_1_2"/><i><font color="Green">E720</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E37:27_6_1_2_1_2"/><i><font color="Green">E721</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, </i>;<br/><a NAME="E38:27_6_1_2_1_2"/><i><font color="Green">E722</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E39:27_6_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><b>hence </b><a NAME="E40:27_6_1_2_1_2"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E102:27"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> holds <br/><span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:27_7"/><i><font color="Green">E723</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 ;<br/>

<a NAME="E2:27_7"/><b>then </b><i><font color="Green">E724</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a></i>;<br/>
<a NAME="E3:27_7"/><i><font color="Green">E725</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a></i>;<br/>
<a NAME="E4:27_7"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<b>hence </b><a NAME="E5:27_7"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, </i>;<br/>


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

<b>thus </b><a NAME="E103:27"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8"><b>proof </b></a><div class="add">

<a NAME="E1:27_8"/><i><font color="Green">E726</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E12:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan7.html#T11" target="_self">Th11</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E2:27_8"/><i><font color="Green">E727</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E3:27_8"/><i><font color="Green">E728</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:27_8"/><i><font color="Green">E729</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> 2
 
<b>by </b><i><a class="ref" href="jordan7.html#T19" target="_self">Th19</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E5:27_8"/><b>then </b><i><font color="Green">E730</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, , <a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:27_8"/><i><font color="Green">E731</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_8"/><i><font color="Green">E732</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E8:27_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:27_8"/><b>then </b><i><font color="Green">E733</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E10:27_8"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E11:27_8"/><b>then </b><i><font color="Green">E734</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E12:27_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/>
<a NAME="E13:27_8"/><b>then </b><i><font color="Green">E735</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E14:27_8"/><i><font color="Green">E736</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:27_8"/><b>then </b><i><font color="Green">E737</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<div><i><font color="Green">E738</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_8_1"/><i><font color="Green">E739</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>
 ;<br/><a NAME="E2:27_8_1"/>
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:27_8_1"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:27_8_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:27_8_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E6:27_8_1"/><b>then </b>
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:27_8_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan7.html#T7" target="_self">Th7</a>, , <a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> implies  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> );<br/>
<a NAME="E17:27_8"/>
<span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E18:27_8"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E19:27_8"/><b>then </b><i><font color="Green">E740</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/>
<a NAME="E20:27_8"/><i><font color="Green">E741</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:27_8_2"/><i><font color="Green">E742</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> implies  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_8_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_8_2_1"/><i><font color="Green">E743</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 ;<br/><a NAME="E2:27_8_2_1"/><i><font color="Green">E744</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:27_8_2_1"/><b>then </b><i><font color="Green">E745</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_8_2_1"/><i><font color="Green">E746</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:27_8_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E6:27_8_2_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E25:27"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, , <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a></i>;<br/><b>hence </b><a NAME="E7:27_8_2_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:27_8_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> implies  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E21:27_8"/><i><font color="Green">E747</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>);<br/></i>
<a NAME="E22:27_8"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E23:27_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
;<br/>
<a NAME="E24:27_8"/><b>then </b>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="jordan7.html#T19" target="_self">Th19</a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E25:27_8"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , </i>;<br/>


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

<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E104:27"/><i><font color="Green">E748</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E105:27"/><i><font color="Green">E749</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E106:27"/><i><font color="Green">E750</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E107:27"/><b>then </b><i><font color="Green">E751</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E108:27"/><i><font color="Green">E752</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a></i>;<br/>
<a NAME="E109:27"/><b>then </b><i><font color="Green">E753</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E110:27"/><b>then </b><i><font color="Green">E754</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E111:27"/><i><font color="Green">E755</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E112:27"/><i><font color="Green">E756</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E113:27"/><i><font color="Green">E757</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, , <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a></i>;<br/>
<a NAME="E114:27"/><i><font color="Green">E758</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a></i>;<br/>
<a NAME="E115:27"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E116:27"/><b>then </b><i><font color="Green">E759</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E117:27"/><i><font color="Green">E760</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E37:27"><i><font color="Green">E106</font></i></a></i>;<br/>
<a NAME="E118:27"/><i><font color="Green">E761</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/>
<a NAME="E119:27"/><i><font color="Green">E762</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E120:27"/><b>then </b><i><font color="Green">E763</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E121:27"/><i><font color="Green">E764</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E122:27"/><i><font color="Green">E765</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green">E766</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_9_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1_1"><b>case </b></a><a NAME="E1:27_9_1_1"/><i><font color="Green">E767</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_9_1_1"/><b>then </b><i><font color="Green">E768</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_9_1_1"/><i><font color="Green">E769</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E47:27"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_9_1_1"/><b>then </b><i><font color="Green">E770</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E43:27"><i><font color="Green">E112</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:27_9_1_1"/><i><font color="Green">E771</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:27_9_1_1"/><i><font color="Green">E772</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E7:27_9_1_1"/><i><font color="Green">E773</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E8:27_9_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E9:27_9_1_1"/><b>then </b>
( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E44:27"><i><font color="Green">E113</font></i></a>, <a class="txt" href="jordan7.html#E46:27"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:27_9_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_9_1_2"><b>case </b></a><a NAME="E1:27_9_1_2"/><i><font color="Green">E774</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_9_1_2"/><b>then </b><i><font color="Green">E775</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E3:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:27_9_1_2"/><b>then </b><i><font color="Green">E776</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E5:27_9_1_2"/><i><font color="Green">E777</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_9_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:27_9_1_2"/><b>then </b><i><font color="Green">E778</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E8:27_9_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E9:27_9_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_9_1_2"/><b>then </b><i><font color="Green">E779</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E11:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E14:27_9_1_2"/><b>then </b><i><font color="Green">E780</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:27_9_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_9_1_2"/><b>then </b><i><font color="Green">E781</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E17:27_9_1_2"/><i><font color="Green">E782</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:27_9_1_2"/><b>then </b><i><font color="Green">E783</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E19:27_9_1_2"/><i><font color="Green">E784</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_9_1_2"/><b>then </b><i><font color="Green">E785</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E21:27_9_1_2"/><i><font color="Green">E786</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E22:27_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E36:27"><i><font color="Green">E105</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E24:27_9_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E25:27_9_1_2"/><b>then </b><i><font color="Green">E787</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E26:27_9_1_2"/><i><font color="Green">E788</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:27_9_1_2"/><i><font color="Green">E789</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E28:27_9_1_2"/><i><font color="Green">E790</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E29:27_9_1_2"/><i><font color="Green">E791</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E30:27_9_1_2"/><i><font color="Green">E792</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_9_1_2"/><i><font color="Green">E793</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_9_1_2"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E33:27_9_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E42:27"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E124:27"/><i><font color="Green">E794</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, </i>;<br/>
<a NAME="E125:27"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E31:27"><i><font color="Green">E100</font></i></a>, </i>;<br/>
<a NAME="E126:27"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, , <a class="txt" href="jordan7.html#E34:27"><i><font color="Green">E103</font></i></a>, <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/>
<b>hence </b><a NAME="E127:27"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, , <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E49:27"><i><font color="Green">E117</font></i></a>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_10"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_10"/><i><font color="Green">E795</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><a NAME="E2:27_10"/><i><font color="Green">E796</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_10"/><b>then </b><i><font color="Green">E797</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:27_10"/><i><font color="Green">E798</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:27_10"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E6:27_10"/><b>then </b><i><font color="Green">E799</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_10_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1_1"><b>suppose </b></a><a NAME="E1:27_10_1_1"/><i><font color="Green">E800</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_10_1_1"/><b>then </b><i><font color="Green">E801</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_10_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E4:27_10_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E32:27"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan7.html#E35:27"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_10_1_2"><b>suppose </b></a><a NAME="E1:27_10_1_2"/><i><font color="Green">E802</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_10_1_2"/><b>then </b><i><font color="Green">E803</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_10_1_2"/><b>then </b><i><font color="Green">E804</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_10_1_2"/><i><font color="Green">E805</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:27_10_1_2"/><i><font color="Green">E806</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_10_1_2"/><b>then </b><i><font color="Green">E807</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_10_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E9:27_10_1_2"/><b>then </b><i><font color="Green">E808</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_10_1_2"/><b>then </b><i><font color="Green">E809</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E11:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_10_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_10_1_2"/><b>then </b><i><font color="Green">E810</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E14:27_10_1_2"/><i><font color="Green">E811</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E15:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_10_1_2"/><b>then </b><i><font color="Green">E812</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:27_10_1_2"/><i><font color="Green">E813</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:27_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E19:27_10_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E20:27_10_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_10_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_10_1_2"/><b>then </b><i><font color="Green">E814</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E23:27_10_1_2"/><i><font color="Green">E815</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_10_1_2"/><b>then </b><i><font color="Green">E816</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_10_1_2"/><i><font color="Green">E817</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E26:27_10_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E33:27"><i><font color="Green">E102</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E27:27_10_1_2"/><b>then </b><i><font color="Green">E818</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E28:27_10_1_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_10_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E40:27"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E30:27_10_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E31:27_10_1_2"/><b>then </b>
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> or <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E32:27_10_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E129:27"/>
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E27:27"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan7.html#E38:27"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan7.html#E39:27"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan7.html#E41:27"><i><font color="Green">E110</font></i></a>, , </i>;<br/>

<b>thus </b><a NAME="E130:27"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> &amp; not <font color="Olive">i</font>,<font color="Olive">j</font> <a href="gobrd10.html#R1">are_adjacent1</a>  holds <br/> <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:27_11"/><i><font color="Green">E819</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:27_11"/><i><font color="Green">E820</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:27_11"/><i><font color="Green">E821</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 <b>and </b><br/><a NAME="E4:27_11"/><i><font color="Green">E822</font></i>: 
not <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="gobrd10.html#R1">are_adjacent1</a> 
 ;<br/>

<a NAME="E5:27_11"/><i><font color="Green">E823</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:27_11"/><i><font color="Green">E824</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:27_11"/><i><font color="Green">E825</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:27_11"/><i><font color="Green">E826</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:27_11"/><i><font color="Green">E827</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a></i>;<br/>
<a NAME="E10:27_11"/><i><font color="Green">E828</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a></i>;<br/>
<a NAME="E11:27_11"/><i><font color="Green">E829</font></i>: 
( not <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; not <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="ref" href="gobrd10.html#D1">GOBRD10:def 1</a></i>;<br/>
<a NAME="E12:27_11"/><i><font color="Green">E830</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E13:27_11"/><b>then </b><i><font color="Green">E831</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E14:27_11"/><i><font color="Green">E832</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:27_11"/><i><font color="Green">E833</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a></i>;<br/>
<a NAME="E16:27_11"/><i><font color="Green">E834</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E17:27_11"/><i><font color="Green">E835</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font> )
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E18:27_11"/><i><font color="Green">E836</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_1_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_1_1_1"/><i><font color="Green">E837</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_1_1_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_1_1_1_1_1"/><i><font color="Green">E838</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_1_1_1_1_1"/><i><font color="Green">E839</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_11_1_1_1_1_1"/><i><font color="Green">E840</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E841</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:27_11_1_1_1_1_1"/><i><font color="Green">E842</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E843</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_11_1_1_1_1_1"/><i><font color="Green">E844</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E845</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_11_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:27_11_1_1_1_1_1"/><b>then </b><i><font color="Green">E846</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:27_11_1_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:27_11_1_1_1_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:27_11_1_1_1_1_1"/><b>then </b>
(  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a>, <a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E14:27_11_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E1:27"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E15:27_11_1_1_1_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_1_1_1_1_2"/><i><font color="Green">E847</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E848</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:27_11_1_1_1_1_2"/><i><font color="Green">E849</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_11_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E850</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E6:27_11_1_1_1_1_2"/><i><font color="Green">E851</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_1_1_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E852</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E9:27_11_1_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:27_11_1_1_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E853</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E12:27_11_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E854</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E15:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E855</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_11_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:27_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E856</font></i>: 
( <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E19:27_11_1_1_1_1_2"/><i><font color="Green">E857</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E858</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E21:27_11_1_1_1_1_2"/><i><font color="Green">E859</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E860</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E23:27_11_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E24:27_11_1_1_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E25:27_11_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E26:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E861</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_1_1_1_1_2"/><i><font color="Green">E862</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E28:27_11_1_1_1_1_2"/><i><font color="Green">E863</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E29:27_11_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E30:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E864</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E31:27_11_1_1_1_1_2"/><i><font color="Green">E865</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E32:27_11_1_1_1_1_2"/><i><font color="Green">E866</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E33:27_11_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E867</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E35:27_11_1_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E36:27_11_1_1_1_1_2"/><b>then </b><i><font color="Green">E868</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E37:27_11_1_1_1_1_2"/><i><font color="Green">E869</font></i>: 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E38:27_11_1_1_1_1_2"/><i><font color="Green">E870</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E39:27_11_1_1_1_1_2"/>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E40:27_11_1_1_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, , , , , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_1_1_2"/><i><font color="Green">E871</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_1_1_2"/><b>then </b><i><font color="Green">E872</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:27_11_1_1_2"/><b>then </b><i><font color="Green">E873</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:27_11_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E5:27_11_1_1_2"/><b>then </b><i><font color="Green">E874</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E6:27_11_1_1_2"/><i><font color="Green">E875</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:27_11_1_1_2"/><b>then </b><i><font color="Green">E876</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E9:27_11_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:27_11_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_1_1_2"/><b>then </b><i><font color="Green">E877</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E12:27_11_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_1_1_2"/><b>then </b><i><font color="Green">E878</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E15:27_11_1_1_2"/><b>then </b><i><font color="Green">E879</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:27_11_1_1_2"/><i><font color="Green">E880</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:27_11_1_1_2"/><i><font color="Green">E881</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:27_11_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_11_1_1_2"/><b>then </b><i><font color="Green">E882</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><i><font color="Green">E883</font></i>: <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E22:27_11_1_1_2"/><b>then </b><i><font color="Green">E884</font></i>: 
( <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> &amp; <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E23:27_11_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:27_11_1_1_2"/><b>then </b><i><font color="Green">E885</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:27_11_1_1_2"/><b>then </b><i><font color="Green">E886</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E26:27_11_1_1_2"/><i><font color="Green">E887</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_1_1_2"/><b>then </b><i><font color="Green">E888</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E28:27_11_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E29:27_11_1_1_2"/><b>then </b><i><font color="Green">E889</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:27_11_1_1_2"/><b>then </b><i><font color="Green">E890</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E31:27_11_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E32:27_11_1_1_2"/><b>then </b><i><font color="Green">E891</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E33:27_11_1_1_2"/><i><font color="Green">E892</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E34:27_11_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E35:27_11_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E36:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E37:27_11_1_1_2"/><b>then </b><i><font color="Green">E893</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E38:27_11_1_1_2"/><i><font color="Green">E894</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><b>set </b><font color="Maroon">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E39:27_11_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E40:27_11_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E41:27_11_1_1_2"/><b>then </b><i><font color="Green">E896</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E42:27_11_1_1_2"/><i><font color="Green">E897</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E43:27_11_1_1_2"/><i><font color="Green">E898</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E44:27_11_1_1_2"/><b>then </b><i><font color="Green">E899</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E45:27_11_1_1_2"/><i><font color="Green">E900</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E46:27_11_1_1_2"/><b>then </b><i><font color="Green">E901</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E47:27_11_1_1_2"/><i><font color="Green">E902</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E48:27_11_1_1_2"/><b>then </b><i><font color="Green">E903</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E49:27_11_1_1_2"/><i><font color="Green">E904</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E50:27_11_1_1_2"/><i><font color="Green">E905</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E51:27_11_1_1_2"/><i><font color="Green">E906</font></i>: 
( <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E52:27_11_1_1_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E53:27_11_1_1_2"/><b>then </b><i><font color="Green">E907</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E54:27_11_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E55:27_11_1_1_2"/><b>then </b><i><font color="Green">E908</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E56:27_11_1_1_2"/><i><font color="Green">E909</font></i>: 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E57:27_11_1_1_2"/><i><font color="Green">E910</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E58:27_11_1_1_2"/><i><font color="Green">E911</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E59:27_11_1_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, , , , , , , , , </i>;<br/><b>hence </b><a NAME="E60:27_11_1_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, , , , , , , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_11_2"/><i><font color="Green">E912</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_2_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1"><b>suppose </b></a><a NAME="E1:27_11_2_1_1"/><i><font color="Green">E913</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_2_1_1"/><i><font color="Green">E914</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_11_2_1_1"/><b>then </b><i><font color="Green">E915</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_2_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_11_2_1_1"/><b>then </b><i><font color="Green">E916</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:27_11_2_1_1"/><i><font color="Green">E917</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E7:27_11_2_1_1"/><b>then </b><i><font color="Green">E918</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E8:27_11_2_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_11_2_1_1"/><b>then </b><i><font color="Green">E919</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_11_2_1_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_11_2_1_1_1_1"/><i><font color="Green">E920</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_2_1_1_1_1"/><b>then </b><i><font color="Green">E921</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_11_2_1_1_1_1"/><i><font color="Green">E922</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_11_2_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:27_11_2_1_1_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E6:27_11_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_11_2_1_1_1_2"/><i><font color="Green">E923</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E924</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_11_2_1_1_1_2"/><i><font color="Green">E925</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:27_11_2_1_1_1_2"/><i><font color="Green">E926</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_11_2_1_1_1_2"/><i><font color="Green">E927</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_11_2_1_1_1_2"/><i><font color="Green">E928</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E929</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_11_2_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E930</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E931</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E12:27_11_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_11_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E932</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E15:27_11_2_1_1_1_2"/><i><font color="Green">E933</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E16:27_11_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E934</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E18:27_11_2_1_1_1_2"/><i><font color="Green">E935</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:27_11_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:27_11_2_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E21:27_11_2_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E22:27_11_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E23:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E936</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:27_11_2_1_1_1_2"/><i><font color="Green">E937</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E938</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E26:27_11_2_1_1_1_2"/><i><font color="Green">E939</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:27_11_2_1_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E28:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E940</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E29:27_11_2_1_1_1_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E30:27_11_2_1_1_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E31:27_11_2_1_1_1_2"/><b>then </b><i><font color="Green">E941</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E32:27_11_2_1_1_1_2"/><i><font color="Green">E942</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E33:27_11_2_1_1_1_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E34:27_11_2_1_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_11_2_1_2"><b>suppose </b></a><a NAME="E1:27_11_2_1_2"/><i><font color="Green">E943</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_11_2_1_2"/><b>then </b><i><font color="Green">E944</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_11_2_1_2"/><b>then </b><i><font color="Green">E945</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E4:27_11_2_1_2"/><i><font color="Green">E946</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E5:27_11_2_1_2"/><i><font color="Green">E947</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_11_2_1_2"/><b>then </b><i><font color="Green">E948</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E8:27_11_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E9:27_11_2_1_2"/><b>then </b><i><font color="Green">E949</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:27_11_2_1_2"/><b>then </b><i><font color="Green">E950</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E11:27_11_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:27_11_2_1_2"/><b>then </b><i><font color="Green">E951</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E14:27_11_2_1_2"/><i><font color="Green">E952</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E15:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E16:27_11_2_1_2"/><b>then </b><i><font color="Green">E953</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E17:27_11_2_1_2"/><i><font color="Green">E954</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_11_2_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E19:27_11_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E20:27_11_2_1_2"/><b>then </b><i><font color="Green">E955</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E21:27_11_2_1_2"/><i><font color="Green">E956</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E22:27_11_2_1_2"/><b>then </b><i><font color="Green">E957</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:27_11_2_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E24:27_11_2_1_2"/><b>then </b><i><font color="Green">E958</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E25:27_11_2_1_2"/><i><font color="Green">E959</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E26:27_11_2_1_2"/><i><font color="Green">E960</font></i>: 
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:27_11_2_1_2"/><b>then </b><i><font color="Green">E961</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E29:27_11_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E30:27_11_2_1_2"/><b>then </b><i><font color="Green">E962</font></i>: 
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E31:27_11_2_1_2"/><b>then </b><i><font color="Green">E963</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E32:27_11_2_1_2"/>
<font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E34:27_11_2_1_2"/><b>then </b><i><font color="Green">E964</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font> )
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="jordan3.html#T5">JORDAN3:5</a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E35:27_11_2_1_2"/><i><font color="Green">E965</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E36:27_11_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E37:27_11_2_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E38:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E39:27_11_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E40:27_11_2_1_2"/><b>then </b><i><font color="Green">E966</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E41:27_11_2_1_2"/><i><font color="Green">E967</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E42:27_11_2_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E43:27_11_2_1_2"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E44:27_11_2_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E20:27_11"/>
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, </i>;<br/>


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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E131:27"/><i><font color="Green">E968</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E132:27"/><i><font color="Green">E969</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 ;<br/>

<a NAME="E133:27"/><i><font color="Green">E970</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E134:27"/><i><font color="Green">E971</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a></i>;<br/>
<a NAME="E135:27"/><i><font color="Green">E972</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E136:27"/><i><font color="Green">E973</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan7.html#E26:27"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a></i>;<br/>
<a NAME="E137:27"/><i><font color="Green">E974</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E53:27"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E138:27"/><b>then </b>
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E139:27"/><b>then </b><i><font color="Green">E975</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, , , <a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan7.html#E62:27"><i><font color="Green">E124</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E140:27"/><i><font color="Green">E976</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E141:27"/><i><font color="Green">E977</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E142:27"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E143:27"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E144:27"/><b>then </b><i><font color="Green">E978</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E145:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/>
<a NAME="E146:27"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E147:27"/><b>then </b><i><font color="Green">E979</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E148:27"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E149:27"/><b>then </b><i><font color="Green">E980</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E22"><i><font color="Green">Lemma78</font></i></a>, , <a class="txt" href="jordan7.html#E65:27"><i><font color="Green">E127</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E150:27"/><i><font color="Green">E981</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E151:27"/><i><font color="Green">E982</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, <a class="txt" href="jordan7.html#E67:27"><i><font color="Green">E129</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E152:27"/><b>then </b><i><font color="Green">E983</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E153:27"/><i><font color="Green">E984</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_12"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_12_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1_1"><b>suppose </b></a><a NAME="E1:27_12_1_1"/><i><font color="Green">E985</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_12_1_1"/><b>then </b><i><font color="Green">E986</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E3:27_12_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:27_12_1_1"/><b>then </b><i><font color="Green">E987</font></i>: 
<font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E5:27_12_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:27_12_1_1"/><b>then </b><i><font color="Green">E988</font></i>: 
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_12_1_1"/><i><font color="Green">E989</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E8:27_12_1_1"/>
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:27_12_1_1"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, , <a class="txt" href="jordan7.html#E71:27"><i><font color="Green">E133</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_12_1_2"><b>suppose </b></a><a NAME="E1:27_12_1_2"/><i><font color="Green">E990</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_12_1_2"/><i><font color="Green">E991</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:27_12_1_2"/><i><font color="Green">E992</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E4:27_12_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:27_12_1_2"/><b>then </b><i><font color="Green">E993</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E6:27_12_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/><a NAME="E7:27_12_1_2"/><b>then </b><i><font color="Green">E994</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E8:27_12_1_2"/><b>then </b><i><font color="Green">E995</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:27_12_1_2"/><i><font color="Green">E996</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:27_12_1_2"/><i><font color="Green">E997</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E11:27_12_1_2"/><i><font color="Green">E998</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E12:27_12_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:27_12_1_2"/><b>then </b><i><font color="Green">E999</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:27_12_1_2"/><b>then </b><i><font color="Green">E1000</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><i><font color="Green">E1001</font></i>: <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>
<br/>.= 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
;<br/><a NAME="E16:27_12_1_2"/><i><font color="Green">E1002</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E17:27_12_1_2"/><i><font color="Green">E1003</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E66:27"><i><font color="Green">E128</font></i></a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:27_12_1_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E19:27_12_1_2"/><b>then </b><i><font color="Green">E1004</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E22"><i><font color="Green">Lemma78</font></i></a>, , <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E20:27_12_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E21:27_12_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E22:27_12_1_2"/><b>then </b><i><font color="Green">E1005</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E23:27_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E24:27_12_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E25:27_12_1_2"/><b>then </b><i><font color="Green">E1006</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E26:27_12_1_2"/><i><font color="Green">E1007</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><b>set </b><font color="Maroon">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E27:27_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:27_12_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:27_12_1_2"/><b>then </b><i><font color="Green">E1008</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E30:27_12_1_2"/><i><font color="Green">E1009</font></i>: 
<font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E31:27_12_1_2"/><i><font color="Green">E1010</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E32:27_12_1_2"/><i><font color="Green">E1011</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E33:27_12_1_2"/><i><font color="Green">E1012</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:27_12_1_2"/><b>then </b><i><font color="Green">E1013</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E35:27_12_1_2"/><i><font color="Green">E1014</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E36:27_12_1_2"/><b>then </b><i><font color="Green">E1015</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E37:27_12_1_2"/><i><font color="Green">E1016</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E38:27_12_1_2"/><i><font color="Green">E1017</font></i>: 
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , , , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E39:27_12_1_2"/><i><font color="Green">E1018</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E64:27"><i><font color="Green">E126</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E40:27_12_1_2"/>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E41:27_12_1_2"/><b>then </b><i><font color="Green">E1019</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E42:27_12_1_2"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan7.html#E68:27"><i><font color="Green">E130</font></i></a>, <a class="txt" href="jordan7.html#E69:27"><i><font color="Green">E131</font></i></a>, <a class="txt" href="jordan7.html#E70:27"><i><font color="Green">E132</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E43:27_12_1_2"/><b>then </b><i><font color="Green">E1020</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E44:27_12_1_2"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan7.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan7.html#E6:27"><i><font color="Green">E37</font></i></a>, <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, , , , , , , , , </i>;<br/><b>hence </b><a NAME="E45:27_12_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_13"/><i><font color="Green">E1021</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_13_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1_1"><b>suppose </b></a><a NAME="E1:27_13_1_1"/><i><font color="Green">E1022</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_13_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:27_13_1_1"/><b>then </b><i><font color="Green">E1023</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:27_13_1_1"/><i><font color="Green">E1024</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h11</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_1.html#T85">FINSEQ_1:85</a></i>;<br/><a NAME="E5:27_13_1_1"/><i><font color="Green">E1025</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E58:27"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:27_13_1_1"/><b>then </b><i><font color="Green">E1026</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E7:27_13_1_1"/>
<font color="Maroon">h1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_13_1_1"/><b>then </b><i><font color="Green">E1027</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E2:27"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan7.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_13_1_1"/><i><font color="Green">E1028</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E10:27_13_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E11:27_13_1_1"/><b>then </b><i><font color="Green">E1029</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E22"><i><font color="Green">Lemma78</font></i></a>, , <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E12:27_13_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E13:27_13_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_13_1_1"/><b>then </b><i><font color="Green">E1030</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E15:27_13_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/><a NAME="E16:27_13_1_1"/><b>then </b><i><font color="Green">E1031</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E17:27_13_1_1"/><i><font color="Green">E1032</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E18:27_13_1_1"/><i><font color="Green">E1033</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/><a NAME="E19:27_13_1_1"/><i><font color="Green">E1034</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:27_13_1_1"/>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E23"><i><font color="Green">Lemma80</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E21:27_13_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E22:27_13_1_1"/><b>then </b><i><font color="Green">E1035</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E23:27_13_1_1"/><i><font color="Green">E1036</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, <a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:27_13_1_1"/><b>then </b><i><font color="Green">E1037</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E25:27_13_1_1"/>
<font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E26:27_13_1_1"/><b>then </b>
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan7.html#E8:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E27:27_13_1_1"/><b>then </b>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E28:27_13_1_1"/><b>then </b><i><font color="Green">E1038</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E29:27_13_1_1"/><i><font color="Green">E1039</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E17:27"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan7.html#E20:27"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E30:27_13_1_1"/>
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E16:27"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan7.html#E18:27"><i><font color="Green">E53</font></i></a>, , <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E31:27_13_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_13_1_2"><b>suppose </b></a><a NAME="E1:27_13_1_2"/><i><font color="Green">E1040</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font>
 ;<br/><div class="add"><a NAME="E2:27_13_1_2"/><b>then </b><i><font color="Green">E1041</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:27_13_1_2"/><i><font color="Green">E1042</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E4:27_13_1_2"/><i><font color="Green">E1043</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E5:27_13_1_2"/><i><font color="Green">E1044</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E74:27"><i><font color="Green">E136</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:27_13_1_2"/><i><font color="Green">E1045</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E76:27"><i><font color="Green">E138</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E7:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:27_13_1_2"/><b>then </b><i><font color="Green">E1046</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, <a class="txt" href="jordan7.html#E77:27"><i><font color="Green">E139</font></i></a>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="txt" href="jordan7.html#E79:27"><i><font color="Green">E141</font></i></a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E9:27_13_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E10:27_13_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:27_13_1_2"/><b>then </b><i><font color="Green">E1047</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E12:27_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , <a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E13:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i>, <a class="txt" href="jordan7.html#E78:27"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:27_13_1_2"/><b>then </b><i><font color="Green">E1048</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E15:27_13_1_2"/><i><font color="Green">E1049</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">j0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E16:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E75:27"><i><font color="Green">E137</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:27_13_1_2"/><b>then </b><i><font color="Green">E1050</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E18:27_13_1_2"/><i><font color="Green">E1051</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="jordan7.html#E81:27"><i><font color="Green">E143</font></i></a>, <a class="txt" href="jordan7.html#E82:27"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:27_13_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E50:27"><i><font color="Green">E118</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E20:27_13_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h0</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E21:27_13_1_2"/><b>then </b><i><font color="Green">E1052</font></i>: 
<font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:27_13_1_2"/><i><font color="Green">E1053</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/><a NAME="E23:27_13_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">h21</font>,2,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan7.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E24:27_13_1_2"/><b>then </b><i><font color="Green">E1054</font></i>: 
<font color="Maroon">h0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h21</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T15" target="_self">Th15</a>, , <a class="txt" href="jordan7.html#E23"><i><font color="Green">Lemma80</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E25:27_13_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E26:27_13_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E27:27_13_1_2"/><b>then </b><i><font color="Green">E1055</font></i>: 
2 <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E28:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , </i>;<br/><a NAME="E29:27_13_1_2"/><b>then </b><i><font color="Green">E1056</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h21</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E30:27_13_1_2"/><i><font color="Green">E1057</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E31:27_13_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E9:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan7.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan7.html#T14" target="_self">Th14</a>, , , , <a class="ref" href="jordan3.html#T1">JORDAN3:1</a></i>;<br/><a NAME="E32:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E51:27"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan7.html#E84:27"><i><font color="Green">E146</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E33:27_13_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E34:27_13_1_2"/><b>then </b><i><font color="Green">E1058</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan7.html#E83:27"><i><font color="Green">E145</font></i></a>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E35:27_13_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h11</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E36:27_13_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan7.html#T17" target="_self">Th17</a>, , <a class="txt" href="jordan7.html#E73:27"><i><font color="Green">E135</font></i></a>, <a class="txt" href="jordan7.html#E80:27"><i><font color="Green">E142</font></i></a>, <a class="txt" href="jordan7.html#E85:27"><i><font color="Green">E147</font></i></a>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E155:27"/>
 <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h0</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h0</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan7.html#E2"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="jordan7.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan7.html#T20" target="_self">Th20</a>, <a class="txt" href="jordan7.html#E55:27"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan7.html#E61:27"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan7.html#E63:27"><i><font color="Green">E125</font></i></a>, <a class="txt" href="jordan7.html#E72:27"><i><font color="Green">E134</font></i></a>, </i>;<br/>


</div>
