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<div class="add">

<b>let </b><font color="Maroon">f</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E45</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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">f</font> )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K33">S-max</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:24"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E6:24"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E7:24"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E8:24"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E9:24"/><i><font color="Green">E59</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E10:24"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E11:24"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E12:24"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E13:24"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E14:24"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:24"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E16:24"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:24"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/>
<a NAME="E18:24"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as  non <a href="realset1.html#V1">trivial</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">i2</font>
;<br/>

<b>take </b>
<font color="Maroon">j2</font>
;<br/>

<a NAME="E20:24"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E21:24"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( ( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/><a NAME="E3:24_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:24_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <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="E5:24_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <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
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:24_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k'</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">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">i3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:24_1_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i3</font>,<font color="Maroon">j3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E9:24_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font>
 <b>and </b><br/><a NAME="E10:24_1_1_2"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i4</font>,<font color="Maroon">j4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E11:24_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i4</font>,<font color="Maroon">j4</font>
 <b>and </b><br/><a NAME="E12:24_1_1_2"/><i><font color="Green">E79</font></i>: 
( ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E13:24_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">k'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E14:24_1_1_2"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> )
 <b>by </b><i>, , <a class="ref" href="jordan1i.html#T3" target="_self">Th3</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E15:24_1_1_2"/><i><font color="Green">E82</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i3</font> &amp; <font color="Maroon">i3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j3</font> &amp; <font color="Maroon">j3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E16:24_1_1_2"/>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:24_1_1_2"/><b>then </b>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:24_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/><a NAME="E19:24_1_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, <a class="ref" href="jordan1i.html#T2" target="_self">Th2</a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><a NAME="E20:24_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">i3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i4</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_2_1_1"/>
( ( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1i.html#E1:24"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan1i.html#T5" target="_self">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_2_1_1_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan1i.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan1i.html#T3" target="_self">Th3</a>, , </i>;<br/><a NAME="E3:24_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1i.html#E5:24"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E4:24_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2_1_1_2"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> )
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1i.html#E5:24"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , , <a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E3:24_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26">S-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="ref" href="jordan1i.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, <a class="ref" href="sprect_2.html#T15">SPRECT_2:15</a></i>;<br/><a NAME="E4:24_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i4</font>,<font color="Maroon">j4</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T118">PSCOMP_1:118</a></i>;<br/><a NAME="E5:24_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">i4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan1i.html#E5:24"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E6:24_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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="24_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_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="E1:24_1_1_2_2_1"/>
( <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> or <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan1i.html#E1:24"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan1i.html#T4" target="_self">Th4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_1"/>
<font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">j3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan1i.html#E5:24"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan1i.html#E7:24"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan1i.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2_2_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:24_1_1_2_2_1_2"/>
<font color="Maroon">k'</font> <a href="nat_1.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>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:24_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, <a class="txt" href="jordan1i.html#E2:24"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a></i>;<br/><a NAME="E4:24_1_1_2_2_1_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font>
 <b>by </b><i/>;<br/><a NAME="E5:24_1_1_2_2_1_2"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k'</font> &amp; <font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 <b>by </b><i>, , , <a class="txt" href="jordan1i.html#E2:24"><i><font color="Green">E45</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><a NAME="E6:24_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, , <a class="ref" href="jordan1i.html#T2" target="_self">Th2</a>, <a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan1i.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan1i.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:24_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:24_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E22:24_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3"><b>suppose </b></a><a NAME="E1:24_1_1_3"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><b>set </b><font color="Maroon">k1</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>;<br/><a NAME="E2:24_1_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</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">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i><a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/><a NAME="E3:24_1_1_3"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:24_1_1_3"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:24_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">k'</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_3.html#T28">FINSEQ_3:28</a></i>;<br/><a NAME="E7:24_1_1_3"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:24_1_1_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:24_1_1_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <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
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:24_1_1_3"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">k'</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">f</font>
 ;<br/><b>consider </b><font color="Maroon">i3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:24_1_1_3"/><i><font color="Green">E98</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i3</font>,<font color="Maroon">j3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E13:24_1_1_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font>
 <b>and </b><br/><a NAME="E14:24_1_1_3"/><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i4</font>,<font color="Maroon">j4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E15:24_1_1_3"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i4</font>,<font color="Maroon">j4</font>
 <b>and </b><br/><a NAME="E16:24_1_1_3"/><i><font color="Green">E103</font></i>: 
( ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> ) or ( <font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E17:24_1_1_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">k'</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">f</font>
 ;<br/><a NAME="E18:24_1_1_3"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> )
 <b>by </b><i>, , , , <a class="txt" href="jordan1i.html#E1:24"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan1i.html#E2:24"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E19:24_1_1_3"/><i><font color="Green">E128</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i3</font> &amp; <font color="Maroon">i3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j3</font> &amp; <font color="Maroon">j3</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="jordan1i.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E20:24_1_1_3"/>
<font color="Maroon">k'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan1i.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:24_1_1_3"/><b>then </b>
<font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E22:24_1_1_3"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#T16">GOBOARD1:16</a></i>;<br/><a NAME="E23:24_1_1_3"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><a NAME="E24:24_1_1_3"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">i3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1_3_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">i3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i4</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_3_1_1"/>
( ( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_3_1_1_1"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:24_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan1i.html#T3" target="_self">Th3</a>, , <a class="txt" href="jordan1i.html#E1:24"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan1i.html#E2:24"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E3:24_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1i.html#E7:24"><i><font color="Green">E49</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><a NAME="E4:24_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_3_1_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> )
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1i.html#E7:24"><i><font color="Green">E49</font></i></a>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K33">S-max</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, , , <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E3:24_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26">S-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> )
 <b>by </b><i>, <a class="ref" href="jordan1i.html#T4" target="_self">Th4</a>, <a class="ref" href="sprect_2.html#T15">SPRECT_2:15</a></i>;<br/><a NAME="E4:24_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i4</font>,<font color="Maroon">j4</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i3</font>,<font color="Maroon">j3</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , , <a class="txt" href="jordan1i.html#E2:24"><i><font color="Green">E45</font></i></a>, <a class="ref" href="pscomp_1.html#T118">PSCOMP_1:118</a></i>;<br/><a NAME="E5:24_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">i4</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan1i.html#E7:24"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E6:24_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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="24_1_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_3_2_1"/>
( <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> or <font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan1i.html#E3:24"><i><font color="Green">E46</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_3_2_1_1"/>
<font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font>
 ;<br/><div class="add"><a NAME="E2:24_1_1_3_2_1_1"/><b>then </b>
<font color="Maroon">j3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j3</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan1i.html#E7:24"><i><font color="Green">E49</font></i></a>, <a class="ref" href="jordan1i.html#T5" target="_self">Th5</a>, , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_3_2_1_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">j3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:24_1_1_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E3:24_1_1_3_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</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">f</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/><b>then </b><i><font color="Green">E136</font></i>: <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k'</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="jordan4.html#T54">JORDAN4:54</a></i>
<br/>.= 
<span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><a NAME="E5:24_1_1_3_2_1_2"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j2</font>
 <b>by </b><i/>;<br/><a NAME="E6:24_1_1_3_2_1_2"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k'</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k'</font> &amp; <font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="txt" href="jordan1i.html#E5:24"><i><font color="Green">E47</font></i></a>, <a class="ref" href="topreal1.html#T27">TOPREAL1:27</a></i>;<br/><a NAME="E7:24_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 <b>by </b><i>, , , <a class="txt" href="jordan1i.html#E6:24"><i><font color="Green">E48</font></i></a>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:24_1_1_3_2_1_2"/><b>then </b>
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E9:24_1_1_3_2_1_2"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:24_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_4"><b>suppose </b></a><a NAME="E1:24_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:24_1_1_4"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="goboard5.html#T5">GOBOARD5:5</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_5"><b>suppose </b></a><a NAME="E1:24_1_1_5"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
<br/>.= 
 <a href="pscomp_1.html#K18">S-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>
;<br/><a NAME="E3:24_1_1_5"/><b>then </b>
( <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K26">S-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f1</font> )
 <b>by </b><i>, , <a class="ref" href="sprect_2.html#T15">SPRECT_2:15</a></i>;<br/><a NAME="E4:24_1_1_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T118">PSCOMP_1:118</a></i>;<br/><a NAME="E5:24_1_1_5"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="goboard5.html#T4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E6:24_1_1_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:24"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 ;<br/>

<b>thus </b><a NAME="E24:24"/>
<font color="Maroon">f</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>by </b><i/>;<br/>


</div>
