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

<b>let </b><font color="Maroon">G</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <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>let </b><font color="Maroon">f2</font> be   <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>assume </b><a NAME="E1:1"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">f1</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,1</span>)</span> &amp; <font color="Maroon">f1</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">f1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span></span>)</span> &amp; <font color="Maroon">f2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,1</span>)</span> &amp; <font color="Maroon">f2</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">f2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font></span>)</span></span>)</span> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">G1</font> being  <a href="goboard1.html#NM1">Go-board</a><br/> for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <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>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Olive">G1</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">g2</font> &amp; <font color="Olive">g1</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Olive">G1</font> &amp; <font color="Olive">g2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Olive">G1</font> &amp; <font color="Olive">g1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Olive">G1</font>,1</span>)</span> &amp; <font color="Olive">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Olive">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">G1</font></span>)</span></span>)</span> &amp; <font color="Olive">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Olive">G1</font>,1</span>)</span> &amp; <font color="Olive">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">g2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Olive">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Olive">G1</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E2:1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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="1_1"><b>proof </b></a><div class="add">

<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><a NAME="E1:1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">G1</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">g1</font> be   <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>let </b><font color="Maroon">g2</font> be   <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>assume </b><a NAME="E2:1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">g2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">g1</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">g1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">g2</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">g2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</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">g1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</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">g2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> );<br/>
<a NAME="E4:1_1"/><i><font color="Green">E50</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">m</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font>;<br/>

<b>thus </b><a NAME="E1:1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<b>thus </b><a NAME="E2:1_1_1"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</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="E6:1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font>;<br/>
<a NAME="E7:1_1"/><i><font color="Green">E53</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E8:1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E9:1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T22">GOBOARD1:22</a></i>;<br/>
<a NAME="E10:1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T23">GOBOARD1:23</a></i>;<br/>
<a NAME="E11:1_1"/>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E12:1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; 0 <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">G1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E13:1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:1_1"/><i><font color="Green">E59</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:1_1"/><i><font color="Green">E60</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E16:1_1"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E17:1_1"/><i><font color="Green">E62</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:1_1"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1, <font color="Maroon">Ll</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> as   <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>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  );<br/>
<a NAME="E20:1_1"/><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</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="E2:1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E3:1_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <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> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<a NAME="E4:1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E5:1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:1_1_2"/>
<font color="Maroon">i</font> <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">G1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E8:1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/>
<a NAME="E9:1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">mi</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="E23:1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">mi</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E24:1_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</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">G1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">ma</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="E26:1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">ma</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> );<br/>
<a NAME="E27:1_1"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E28:1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>]
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/>
<a NAME="E29:1_1"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">ma1</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="E31:1_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">ma1</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i>
<a NAME="E32:1_1"/><i><font color="Green">E81</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E34:1_1"/><i><font color="Green">E83</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_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="E1:1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:1_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:1_1_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">Lmi</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font> as   <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/>
<a NAME="E36:1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> implies <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/>

<b>consider </b><font color="Maroon">n</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="E3:1_1_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">i</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="E5:1_1_4"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_4"/><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ci</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font> as   <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/>
<a NAME="E8:1_1_4"/><i><font color="Green">E92</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E9:1_1_4"/><b>then </b>
( 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>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</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="E11:1_1_4"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#T49">GOBOARD1:49</a></i>;<br/>
<i><font color="Green">E94</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i</font></span>)</span> = 
 <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">G1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">j</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="E14:1_1_4"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ci</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ci</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E15:1_1_4"/><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ci</font> <a href="hidden.html#R1">=</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">Ci</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Lj</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">j</font> as   <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/>
<a NAME="E17:1_1_4"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lj</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E20:1_1_4"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lj</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E101</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> = 
<font color="Maroon">Ci</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/>.= 
<font color="Maroon">Lj</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>
<br/>.= 
<font color="Maroon">Lj</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E22:1_1_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lj</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E23:1_1_4"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lj</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Lj</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E25:1_1_4"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_4_4"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">mi</font>
 ;<br/><a NAME="E2:1_1_4_4"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">mi</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:1_1_4_4"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="goboard4.html#T1" target="_self">Th1</a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E27:1_1_4"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>, , <a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k1</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="E38:1_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lmi</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E39:1_1"/><i><font color="Green">E105</font></i>: 
 <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">Lmi</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lmi</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ck1</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k1</font> as   <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/>
<a NAME="E41:1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</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">l1</font> = <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1, <font color="Maroon">l2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">ma</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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> = <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">k1</font>;<br/>
<b>consider </b><font color="Maroon">h1</font> being   <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><b> such that </b><br/><a NAME="E44:1_1"/><i><font color="Green">E111</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> holds <br/><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i>
<a NAME="E45:1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c1</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1 as   <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>consider </b><font color="Maroon">x</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="E4:1_1_5_1_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_1"/><i><font color="Green">E116</font></i>: 
(  <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">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c1</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</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">G1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>reconsider </b><font color="Maroon">lx</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">x</font> as   <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/><a NAME="E7:1_1_5_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E8:1_1_5_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E9:1_1_5_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp;  <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">lx</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lx</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lx</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp;  <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">g2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E10:1_1_5_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon">m</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="E12:1_1_5_1_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">lx</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T45">GOBOARD1:45</a></i>;<br/><b>consider </b><font color="Maroon">y</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="E14:1_1_5_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lx</font> &amp; <font color="Maroon">lx</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="goboard4.html#T1" target="_self">Th1</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E15:1_1_5_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">lx</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">c1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E16:1_1_5_1_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E17:1_1_5_1_1"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E18:1_1_5_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="goboard4.html#T1" target="_self">Th1</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E19:1_1_5_1_1"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E20:1_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_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="E1:1_1_5_1_2_1_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_1"/>
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">ma1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font></span>)</span> );<br/><a NAME="E2:1_1_5_1_2_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#T53">GOBOARD1:53</a></i>;<br/><b>consider </b><font color="Maroon">mi1</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="E6:1_1_5_1_2_1_1_2"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">mi1</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">mi1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="goboard4.html#E1:1"><i><font color="Green">E43</font></i></a>);<br/></i><b>set </b><font color="Maroon">f1</font> = <font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font>;<br/><a NAME="E7:1_1_5_1_2_1_1_2"/>
( <font color="Maroon">ma1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> &amp; <font color="Maroon">mi1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2"/><b>then </b>
<font color="Maroon">ma1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> <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/><b>then reconsider </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">mi1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ma1</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</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">w</font> = <font color="Maroon">ma1</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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</font> <a href="real_1.html#K5">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E137</font></i>: 
( 0 <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">ma1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma1</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mi1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E13:1_1_5_1_2_1_1_2"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> holds <br/> ex <font color="Olive">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> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">n</font>,<font color="Olive">p</font>]
 ;<br/><b>consider </b><font color="Maroon">f</font> being   <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><b> such that </b><br/><a NAME="E15:1_1_5_1_2_1_1_2"/><i><font color="Green">E139</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E3:1"><i><font color="Green">E46</font></i></a>);<br/></i><a NAME="E16:1_1_5_1_2_1_1_2"/><i><font color="Green">E140</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">Lma</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font> as   <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>consider </b><font color="Maroon">k2</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="E19:1_1_5_1_2_1_1_2"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lma</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k2</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2"/><i><font color="Green">E144</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lma</font> <a href="hidden.html#R1">=</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">Lma</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">Ck2</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k2</font> as   <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/><a NAME="E22:1_1_5_1_2_1_1_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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> = <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">k2</font>;<br/><b>consider </b><font color="Maroon">h2</font> being   <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><b> such that </b><br/><a NAME="E24:1_1_5_1_2_1_1_2"/><i><font color="Green">E148</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> holds <br/><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E25:1_1_5_1_2_1_1_2"/><i><font color="Green">E149</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</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">h2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">ff</font> = <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>;<br/><a NAME="E26:1_1_5_1_2_1_1_2"/>
( <font color="Maroon">ma1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> &amp; <font color="Maroon">mi1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</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="E27:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
( <font color="Maroon">ma1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <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">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  <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">h2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</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="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E28:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E29:1_1_5_1_2_1_1_2"/><i><font color="Green">E153</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green">E154</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_2"/><b>then </b><i><font color="Green">E155</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E156</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_3"/>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_4"/><b>then </b><i><font color="Green">E158</font></i>: 
( <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">k1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E57</font></i></a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">n</font>;<br/><b>take </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">k1</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E159</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E160</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k2</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><b>take </b><font color="Maroon">k2</font> = <font color="Maroon">k2</font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">k2</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">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E161</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_6"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E162</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_6"/><b>then </b>
( <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E5:1_1_5_1_2_1_1_2_6"/><i><font color="Green">E163</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E6:1_1_5_1_2_1_1_2_6"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E17:1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E14:1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E165</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_7"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">n</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">h1</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E166</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">k1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E167</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">k1</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k1</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">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_7"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_7"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E38:1_1_5_1_2_1_1_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi1</font>
 ;<br/><div><i><font color="Green">E173</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_8"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">n</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">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green">E174</font></i>: 
( <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">k2</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_8"/><b>then </b><i><font color="Green">E175</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">k2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k2</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">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_8"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_8"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E176</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_9"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">n</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> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_9"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_9"/><b>then </b><i><font color="Green">E177</font></i>: 
( <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <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">g1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_9"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_9"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E178</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E179</font></i>: 
( <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">k1</font> &amp; <font color="Maroon">l1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_10"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">k1</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E180</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_10"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma1</font>
 ;<br/><a NAME="E7:1_1_5_1_2_1_1_2_10"/><b>then </b>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><b>then </b><i><font color="Green">E183</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, </i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">mi</font>,<font color="Maroon">k1</font>
<b>by </b><i>, , <a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_10"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">mi</font>,<font color="Maroon">k1</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">G1</font>
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_10"/><b>then </b>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E42:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> holds <br/>for <font color="Olive">i1</font>, <font color="Olive">i2</font>, <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j1</font>,<font color="Olive">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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">j1</font>,<font color="Olive">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="real_1.html#K5">-</a> <font color="Olive">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="goboard4.html#E18:1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon">hf</font> = <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E186</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E187</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k2</font> &amp; <font color="Maroon">ma</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">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_11"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k2</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">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E188</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E189</font></i>: 
(  <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> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_11"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E191</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, </i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ma</font>,<font color="Maroon">k2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_11"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ma</font>,<font color="Maroon">k2</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_11"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k2</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E44:1_1_5_1_2_1_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><a NAME="E45:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E17:1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green">E193</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_12_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E196</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</font> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_12_1_1"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">ma1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mi1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_12_1_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma1</font>
 ;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, </i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_12_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_12_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_12_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E200</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E201</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_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="E7:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E202</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_12_1_2"/><i><font color="Green">E203</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_12_1_2"/>
0 <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><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_12_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><i><font color="Green">E205</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">k1</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E15:1_1_5_1_2_1_1_2_12_1_2"/>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E16:1_1_5_1_2_1_1_2_12_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E206</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E48:1_1_5_1_2_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 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">h1</font></span>)</span> <a href="nat_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>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E49:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E207</font></i>: 
0 <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">f</font></span>)</span> <a href="nat_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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E208</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_14_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_14_1_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b>
<font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b><i><font color="Green">E210</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_14_1_1"/><b>then </b><i><font color="Green">E211</font></i>: 
(  <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> &amp; <font color="Maroon">mi1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mi1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font>
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, , <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, , <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi1</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_14_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_14_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">ma</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">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E213</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E214</font></i>: 
<font color="Maroon">l2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E215</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_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="E8:1_1_5_1_2_1_1_2_14_1_2"/><i><font color="Green">E216</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_14_1_2"/><b>then </b><i><font color="Green">E217</font></i>: 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><i><font color="Green">E218</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l2</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span>,<font color="Maroon">k2</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E7:1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E12:1_1_5_1_2_1_1_2_14_1_2"/>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:1_1_5_1_2_1_1_2_14_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E219</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E52:1_1_5_1_2_1_1_2"/><i><font color="Green">E220</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<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="E1:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E222</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">gg</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E224</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma1</font>
 
;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_16"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E226</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_16"/><i><font color="Green">E227</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E228</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><i><font color="Green">E231</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k1</font></span>)</span> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E232</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ck1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_1"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">Ck1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">mi</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_16_2_1_1"/>
<font color="Maroon">Ck1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck1</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="E11:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, , , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ck1</font>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b><i><font color="Green">E234</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">Ck1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font>
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_16_2_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_16_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_16_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_16_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E235</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">k2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><i><font color="Green">E238</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k2</font></span>)</span> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E239</font></i>: 
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ck2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_16_2_1_2"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k2</font> &amp; <font color="Maroon">Ck2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_16_2_1_2"/>
<font color="Maroon">Ck2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, , , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ck2</font>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b><i><font color="Green">E241</font></i>: 
( <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ck2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E3:1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_16_2_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_16_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:1_1_5_1_2_1_1_2_16"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E53:1_1_5_1_2_1_1_2"/><i><font color="Green">E242</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_17"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E243</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E244</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</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="E5:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E246</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n1</font>,<font color="Maroon">k1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E8:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E249</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<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">i2</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="E10:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E250</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font> as   <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/>
<a NAME="E12:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E252</font></i>: 
 <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">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_17"/><i><font color="Green">E254</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_17"/><b>then </b>
<font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E17:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_17"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_17"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_17"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">k1</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">G1</font>
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_17"/><b>then </b>
( <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_17"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E54:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_18"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E256</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E257</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</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="E5:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E258</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E259</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>,<font color="Maroon">k2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</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="E8:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<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">i2</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="E10:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E261</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font> as   <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/>
<a NAME="E12:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E262</font></i>: 
 <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">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E263</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_18"/><i><font color="Green">E264</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E17:1_1_5_1_2_1_1_2_18"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E18:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_18"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_18"/>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E13:1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_18"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>,<font color="Maroon">k2</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">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_18"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_18"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E266</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E26:1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>

;<br/><a NAME="E56:1_1_5_1_2_1_1_2"/><i><font color="Green">E267</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_20"><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="E1:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:1_1_5_1_2_1_1_2_20"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_20"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_20"/><b>then </b>
( <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_20"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E57:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E269</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E27:1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E58:1_1_5_1_2_1_1_2"/><i><font color="Green">E270</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E59:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E271</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D10">GOBOARD1:def 10</a></i>;<br/><a NAME="E60:1_1_5_1_2_1_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E273</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>thus </b><a NAME="E62:1_1_5_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E274</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> );<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E275</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E276</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">m</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[1]
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T22">GOBOARD1:22</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">m</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="E7:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E277</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">m</font>] &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S6">NAT_1:sch 6</a>(, );<br/></i><b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</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/><div><i><font color="Green">E278</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_2"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">ll</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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> = <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><b>consider </b><font color="Maroon">t</font> being   <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><b> such that </b><br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E282</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ll</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E283</font></i>: 
 <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">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E284</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ll</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E285</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> holds <br/><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ll</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_3"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E286</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</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="E3:1_1_5_1_2_1_1_2_22_1_1_4"/><i><font color="Green">E287</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ll</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <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>set </b><font color="Maroon">D</font> =  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E289</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1"/>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E290</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_1"/>
<font color="Maroon">m</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> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E291</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ll</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E25:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E292</font></i>: 
<font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_1"/>
(  <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <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="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T30">GOBOARD1:30</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E293</font></i>: 
<font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>,1</span>)</span>
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, , <a class="ref" href="goboard1.html#T48">GOBOARD1:48</a></i>;<br/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">ll</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><font color="Maroon">t</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">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ll</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>

;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">t</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">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#T30">GOBOARD1:30</a></i>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_1"/><i><font color="Green">E295</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><div><i><font color="Green">E296</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><b>then 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">i2</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="E4:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E298</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">i1</font> = <font color="Maroon">i1</font>;<br/><b>reconsider </b><font color="Maroon">Ci2</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i2</font> as   <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/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_6"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_6"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ci2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green">E301</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ci2</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">Ci2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ci2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
( <font color="Maroon">Ci2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E302</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_6_2"/>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b><i><font color="Green">E303</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, , , , <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_6_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_6_2"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E13:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E304</font></i>: 
( 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i2</font> &amp; 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="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E305</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">i2</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="int_1.html#T18">INT_1:18</a></i>;<br/><b>take </b><font color="Maroon">l</font> = <font color="Maroon">l</font>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_1_6"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b><i><font color="Green">E306</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_1_6"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E19:1_1_5_1_2_1_1_2_22_1_1_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">l</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b><a NAME="E20:1_1_5_1_2_1_1_2_22_1_1_6"/>
<font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">l</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="goboard1.html#T32">GOBOARD1:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_1_7"/><i><font color="Green">E307</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">t</font> &amp; <font color="Maroon">n</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">t</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E308</font></i>: 
( <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_1_7"/><i><font color="Green">E309</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E310</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b>
( 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> <font color="Maroon">k</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> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E311</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></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="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i2</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="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <font color="Maroon">j2</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="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, , , <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="goboard1.html#T32">GOBOARD1:32</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_1_7"/><b>then </b><i><font color="Green">E312</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_1_7"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a></i>;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span>

;<br/><br/></div><b>end;</b></div><a NAME="E34:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E313</font></i>: 
<font color="Maroon">t</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E35:1_1_5_1_2_1_1_2_22_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , , , <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, , <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b><i><font color="Green">E314</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, , , , , , </i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E40:1_1_5_1_2_1_1_2_22_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2"/><i><font color="Green">E315</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">D</font> =  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2"/><i><font color="Green">E316</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b><i><font color="Green">E317</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon">t</font> being   <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><b> such that </b><br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><i><font color="Green">E318</font></i>: 
( <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon">t</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">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#T51">GOBOARD1:51</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><i><font color="Green">E319</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="txt" href="goboard4.html#E23:1_1"><i><font color="Green">E72</font></i></a>, , <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E21:1_1"><i><font color="Green">E70</font></i></a>, , <a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1"><i><font color="Green">E78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E321</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E322</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <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><b> such that </b><br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E323</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">t</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">t</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">t</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E34:1_1"><i><font color="Green">E83</font></i></a>, , <a class="ref" href="goboard1.html#T52">GOBOARD1:52</a></i>;<br/><b>consider </b><font color="Maroon">tt</font> being   <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><b> such that </b><br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E325</font></i>: 
( <font color="Maroon">tt</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <font color="Maroon">tt</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">tt</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font> &amp; <font color="Maroon">tt</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T51">GOBOARD1:51</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> );<br/><div><i><font color="Green">E326</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/><i><font color="Green">E327</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font></span>)</span> );<br/><div><i><font color="Green">E329</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font></span>)</span></span>)</span>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">n</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/><i><font color="Green">E330</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>take </b><font color="Maroon">n</font> = <font color="Maroon">n</font>;<br/><b>thus </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">n</font>]
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">pf</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="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">pf</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>consider </b><font color="Maroon">pl</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="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon">pl</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E335</font></i>: 
<font color="Maroon">pl</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">pf</font>
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E336</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> &amp; <font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> &amp; <font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1"/>
( <font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pl</font> or <font color="Maroon">pf</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">pl</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/>
<font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pl</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/><b>then </b>
<font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_1"/>
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> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/>
<font color="Maroon">pf</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">pl</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/><b>then </b>
<font color="Maroon">pl</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_3_1_2"/>
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> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <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><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E337</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> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">lg</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
<font color="Maroon">lg</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E339</font></i>: 
( <font color="Maroon">lg</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">lg</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>, <font color="Maroon">Li</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font>, <font color="Maroon">La</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">ma</font> as   <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/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E343</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><b>then </b><i><font color="Green">E344</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>consider </b><font color="Maroon">K1</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="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E346</font></i>: 
( <font color="Maroon">K1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Li</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E347</font></i>: 
 <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">Li</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Li</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">CK1</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K1</font> as   <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/><a NAME="E28:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E349</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Li</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E350</font></i>: 
<font color="Maroon">Li</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E351</font></i>: 
( <font color="Maroon">K1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">mi</font>,<font color="Maroon">K1</font> )
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E352</font></i>: 
<font color="Maroon">pf</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">i2</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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E353</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E354</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E355</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">mi</font>,<font color="Maroon">K1</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><b>then </b><i><font color="Green">E356</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</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="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E357</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E358</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/><i><font color="Green">E359</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> ) )
 </span><b>by </b><i>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><i><font color="Green">E360</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 ;<br/><div class="add"><i><font color="Green">E361</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> = 
 <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">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> )
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lg</font> &amp; <font color="Maroon">lg</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><i><font color="Green">E362</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Li</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b><i><font color="Green">E363</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Li</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
<font color="Maroon">Li</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Li</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lg</font> &amp; <font color="Maroon">lg</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_5"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E364</font></i>: 
<font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">K2</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="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E366</font></i>: 
( <font color="Maroon">K2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lma</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E367</font></i>: 
 <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">Lma</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lma</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">CK2</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K2</font> as   <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/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E369</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lma</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E370</font></i>: 
<font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E371</font></i>: 
( <font color="Maroon">K2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ma</font>,<font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E372</font></i>: 
<font color="Maroon">pl</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">i2</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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E373</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><i><font color="Green">E374</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> = 
 <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">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E375</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E376</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ma</font>,<font color="Maroon">K2</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><b>then </b><i><font color="Green">E377</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</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="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E378</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E379</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/><i><font color="Green">E380</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1"/>
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> ) )
 </span><b>by </b><i>, <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><i><font color="Green">E381</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> )
 ;<br/><div class="add"><i><font color="Green">E382</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">C</font> = 
 <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">C</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> )
 <b>by </b><i>, , , , , , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lg</font> &amp; <font color="Maroon">lg</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><i><font color="Green">E383</font></i>: 
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lma</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b><i><font color="Green">E384</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lma</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
<font color="Maroon">La</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">lg</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Lma</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lg</font> &amp; <font color="Maroon">lg</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/><b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_6"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><b>then </b><i><font color="Green">E385</font></i>: 
<font color="Maroon">pl</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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> = <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">K1</font>;<br/><b>consider </b><font color="Maroon">h1</font> being   <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><b> such that </b><br/><a NAME="E43:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E386</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> holds <br/><font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E387</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="finseq_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/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</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> = <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">K2</font>;<br/><b>consider </b><font color="Maroon">h2</font> being   <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><b> such that </b><br/><a NAME="E46:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E388</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l2</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> holds <br/><font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">n</font>) ) )
 <b>from </b><i><a class="ref" href="goboard2.html#S1">GOBOARD2:sch 1</a>();<br/></i><a NAME="E47:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E389</font></i>: 
 <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">h2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><i><font color="Green">E390</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b><i><font color="Green">E391</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">l1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T45">XREAL_1:45</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_7"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E392</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_8"/>
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E393</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/><b>then </b><i><font color="Green">E394</font></i>: 
( <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">K1</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, </i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">n</font>;<br/><b>take </b><font color="Maroon">K1</font> = <font color="Maroon">K1</font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_9"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">K1</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">K1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E395</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/><b>then </b><i><font color="Green">E396</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>;<br/><b>take </b><font color="Maroon">K2</font> = <font color="Maroon">K2</font>;<br/><b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_10"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">K2</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">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E397</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">n</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">h1</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E398</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">K1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b><i><font color="Green">E399</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">K1</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K1</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">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_11"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><div><i><font color="Green">E400</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">n</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">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b><i><font color="Green">E401</font></i>: 
( <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b><i><font color="Green">E402</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">K2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K2</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">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_12"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>9</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">m</font></span>)</span> holds <br/><font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> );<br/><a NAME="E54:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E404</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E55:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E405</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>9</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"><b>proof </b></a><div class="add">

<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><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E406</font></i>: 
<font color="Maroon">S<sub>9</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E407</font></i>: 
<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>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/><i><font color="Green">E408</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">m</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><i><font color="Green">E409</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><i><font color="Green">E410</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">ma1</font>
 ;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/><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>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Li</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_1"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &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>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E411</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <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><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E412</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/><b>then 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">i2</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E413</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">Li1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font> as   <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/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E415</font></i>: 
 <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">Li1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Li1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E416</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E417</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Li1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E418</font></i>: 
<font color="Maroon">Li1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
<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">Li1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E419</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="relat_1.html#K2">rng</a> <font color="Maroon">Li1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E420</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E126</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><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">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="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E421</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</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#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">Lj1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">j1</font> as   <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/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E423</font></i>: 
 <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">Lj1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lj1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E424</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><i><font color="Green">E425</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lj1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b>
<font color="Maroon">Lj1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lj1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_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">Lj1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/><b>then </b><i><font color="Green">E426</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="relat_1.html#K2">rng</a> <font color="Maroon">Lj1</font>
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> or <font color="Maroon">mi</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E427</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E428</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> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a></i>;<br/><font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a></i>
<br/>.= 
<font color="Maroon">Li1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E429</font></i>: 
<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">mi</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma1</font> &amp; <font color="Maroon">ma1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E430</font></i>: 
<font color="Maroon">ma1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><i><font color="Green">E431</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> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</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="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, </i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
( <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="goboard4.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mi</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">G1</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E432</font></i>: 
<font color="Maroon">mi</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">G1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><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>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T44">GOBOARD1:44</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, , <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_1"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1"/>
( <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font></span>)</span> or for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">l</font></span>)</span> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, , <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="goboard1.html#T43">GOBOARD1:43</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"/>
<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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">l</font></span>)</span> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E434</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, , <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">mi</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, , <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2_1_1_2"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E25:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13_1_1_2"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_13"/>
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E56:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E435</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>9</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E57:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E436</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E437</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E438</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">i1</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E439</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</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="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E440</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</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">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ln1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">n1</font> as   <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/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E442</font></i>: 
 <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">Ln1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ln1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E443</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E444</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ln1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">Ln1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ln1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ln1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ln1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b><i><font color="Green">E445</font></i>: 
<font color="Maroon">mi</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">i2</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="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E446</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E126</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E447</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="finseq_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="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><i><font color="Green">E448</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">K1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, , </i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">K1</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">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/><b>then </b><i><font color="Green">E449</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
( <font color="Maroon">l1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_14"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>10</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> implies for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">m</font></span>)</span> holds <br/><font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font> );<br/><a NAME="E58:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E450</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[0]
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E59:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E451</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>10</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>10</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"><b>proof </b></a><div class="add">

<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><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E452</font></i>: 
<font color="Maroon">S<sub>10</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E453</font></i>: 
<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>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/><i><font color="Green">E454</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">m</font></span>)</span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><i><font color="Green">E455</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E9:1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/><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>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Li</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E456</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &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>by </b><i><a class="txt" href="goboard4.html#E8:1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E457</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <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><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E458</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/><b>then 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">i2</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E459</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">Li1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font> as   <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/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E460</font></i>: 
 <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">Li1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Li1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E461</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E462</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Li1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">Li1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Li1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<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">Li1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E463</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="relat_1.html#K2">rng</a> <font color="Maroon">Li1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E464</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, , <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>consider </b><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">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="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E465</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</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#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">Lj1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">j1</font> as   <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/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E466</font></i>: 
 <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">Lj1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Lj1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E467</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><i><font color="Green">E468</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Lj1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b>
<font color="Maroon">Lj1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lj1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_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">Lj1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E469</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="relat_1.html#K2">rng</a> <font color="Maroon">Lj1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/><b>then </b><i><font color="Green">E470</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, , <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ma</font> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E471</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E472</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> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">w</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g1</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#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">mi1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</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="finseq_1.html#K4">dom</a> <font color="Maroon">g1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E2:1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><i><font color="Green">E473</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
<font color="Maroon">ma1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , , , , <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">ma1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E474</font></i>: 
<font color="Maroon">mi1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E1:1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"/><i><font color="Green">E475</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">ma</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1"/>
( <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font></span>)</span> or for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">l</font></span>)</span> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E36:1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, , <a class="ref" href="goboard1.html#T43">GOBOARD1:43</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"/>
<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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">i1</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, , <a class="ref" href="goboard1.html#T19">GOBOARD1:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/>
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Olive">l</font></span>)</span> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E476</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, , <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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_15"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E60:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E477</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>10</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="goboard4.html#E1:1_1_5_1_2"><i><font color="Green">E125</font></i></a>, <a class="txt" href="goboard4.html#E2:1_1_5_1_2"><i><font color="Green">E126</font></i></a>);<br/></i><a NAME="E61:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/><i><font color="Green">E478</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E479</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E480</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">i1</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E481</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</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="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E482</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</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">G1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n1</font>,<font color="Maroon">n2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1"><i><font color="Green">E61</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Ln1</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<font color="Maroon">n1</font> as   <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/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E483</font></i>: 
 <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">Ln1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ln1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E484</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E485</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Ln1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">Ln1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ln1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ln1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ln1</font>
 
<b>by </b><i>, , <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b><i><font color="Green">E486</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ma</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E1:1_1_5_1_2_1_1_2"><i><font color="Green">E127</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">i2</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="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E487</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><i><font color="Green">E488</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">K2</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">G1</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E21:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b><i><font color="Green">E489</font></i>: 
( <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_16"/>
contradiction
 <b>by </b><i>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1"/>
( <font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pl</font> or <font color="Maroon">pl</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pf</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="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E490</font></i>: 
<font color="Maroon">pf</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pl</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">F1</font> = <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
( <font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> &amp; <font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
<font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> <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/><b>then reconsider </b><font color="Maroon">LL</font> = <span class="p1">(<span class="default"><font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">pf</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/><b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">pf</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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E494</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font></span>)</span> &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><i><font color="Green">E495</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b><i><font color="Green">E496</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">pf</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">pf</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pl</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E497</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/> ex <font color="Olive">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> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive">n</font>,<font color="Olive">p</font>]
 ;<br/><b>consider </b><font color="Maroon">F</font> being   <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><b> such that </b><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E498</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LL</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>();<br/></i><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E499</font></i>: 
 <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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">FF</font> = <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
( <font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> &amp; <font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
( <font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E501</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LL</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E502</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><b>then </b><i><font color="Green">E503</font></i>: 
( <font color="Maroon">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/><i><font color="Green">E504</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, </i>;<br/></div><b>end;</b></div><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E505</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E506</font></i>: 
<font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pl</font>
 ;<br/><div><i><font color="Green">E507</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</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> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/><b>then </b><i><font color="Green">E508</font></i>: 
( <font color="Maroon">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E509</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><i><font color="Green">E510</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b><i><font color="Green">E511</font></i>: 
( <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">K1</font> &amp; <font color="Maroon">l1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">K1</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">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b><i><font color="Green">E512</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/>
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a></i>;<br/><b>then </b><i><font color="Green">E513</font></i>: <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">mi</font>,<font color="Maroon">K1</font>
<b>by </b><i>, , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">mi</font>,<font color="Maroon">K1</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_4"/><b>then </b>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><b>set </b><font color="Maroon">hf</font> = <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font>;<br/><div><i><font color="Green">E514</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><i><font color="Green">E515</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b><i><font color="Green">E516</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">ma</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">G1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K2</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">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b><i><font color="Green">E517</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><i><font color="Green">E518</font></i>: 
(  <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> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><i><font color="Green">E519</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ma</font>,<font color="Maroon">K2</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ma</font>,<font color="Maroon">K2</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_5"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E520</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a><br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> holds <br/>for <font color="Olive">i1</font>, <font color="Olive">i2</font>, <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j1</font>,<font color="Olive">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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">j1</font>,<font color="Olive">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="real_1.html#K5">-</a> <font color="Olive">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_6"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E521</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E522</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><b>then </b><i><font color="Green">E523</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E524</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/><i><font color="Green">E525</font></i>: 
<font color="Maroon">pf</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pl</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><i><font color="Green">E526</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E527</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E528</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><i><font color="Green">E529</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E530</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b><i><font color="Green">E531</font></i>: 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K1</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
0 <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><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">K1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font>
<b>by </b><i>, , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_7_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E532</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
0 <a href="nat_1.html#K1">+</a> 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">h1</font></span>)</span> <a href="nat_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>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E24:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E533</font></i>: 
0 <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">F</font></span>)</span> <a href="nat_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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E534</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><i><font color="Green">E535</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b>
<font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b><i><font color="Green">E536</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/><b>then </b><i><font color="Green">E537</font></i>: 
(  <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> &amp; <font color="Maroon">pl</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pl</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, , <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font>
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><i><font color="Green">E538</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b>
<font color="Maroon">ma</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">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E539</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E540</font></i>: 
<font color="Maroon">l2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E541</font></i>: 
<font color="Maroon">K2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><b>then </b><i><font color="Green">E542</font></i>: 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/><i><font color="Green">E543</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span>,<font color="Maroon">K2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_9_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E544</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E545</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"><b>proof </b></a><div class="add">

<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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E546</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E547</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">gg</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E548</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E549</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/><i><font color="Green">E550</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E551</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E552</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E553</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">K1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><i><font color="Green">E554</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK1</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E555</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><i><font color="Green">E556</font></i>: 
( <font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">mi</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
<font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">CK1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b><i><font color="Green">E557</font></i>: 
( <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> )
 <b>by </b><i>, , , , , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</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">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font>
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , , , <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E558</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E559</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E560</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">K2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><i><font color="Green">E561</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK2</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E562</font></i>: 
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E563</font></i>: 
( <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
<font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">CK2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b><i><font color="Green">E564</font></i>: 
<font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><i><font color="Green">E565</font></i>: 
<font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font></span>)</span> )
 <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , , , <a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_11"/>
contradiction
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E566</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>

;<br/><a NAME="E29:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E567</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E31:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"><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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><i><font color="Green">E568</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><b>then </b>
( <font color="Maroon">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E569</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><i><font color="Green">E570</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E571</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E572</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</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="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E573</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><i><font color="Green">E574</font></i>: 
 <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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b><i><font color="Green">E575</font></i>: 
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> &amp; <font color="Maroon">w1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pl</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> &amp; <font color="Maroon">w1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/>
( <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> &amp; <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font> )
 
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b><i><font color="Green">E576</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/><b>then </b>
<font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">LL</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1_14"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E577</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/><b>then </b><i><font color="Green">E578</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a></i>;<br/><b>hence </b><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E579</font></i>: 
<font color="Maroon">pl</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">pf</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">F1</font> = <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font>;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
( <font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> &amp; <font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
<font color="Maroon">pl</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <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/><b>then reconsider </b><font color="Maroon">LL</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">pl</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E580</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 ;<br/>

<a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b><i><font color="Green">E581</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b>
<font color="Maroon">LL</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <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/>
<b>hence </b><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E582</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Olive">n</font> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E583</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E584</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E585</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">LL</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <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/>
<b>then reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> 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="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><i><font color="Green">E586</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">LL</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pf</font>
 
<b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>11</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E587</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/> ex <font color="Olive">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> st <font color="Maroon">S<sub>11</sub></font>[<font color="Olive">n</font>,<font color="Olive">p</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/><i><font color="Green">E588</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a></i>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_3"/>
<font color="Maroon">S<sub>11</sub></font>[<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">F</font> being   <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><b> such that </b><br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E589</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LL</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> holds <br/><font color="Maroon">S<sub>11</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>] ) )
 <b>from </b><i><a class="ref" href="goboard1.html#S1">GOBOARD1:sch 1</a>(<a class="txt" href="goboard4.html#E6:1_1_5_1_2_1_1_2"><i><font color="Green">E132</font></i></a>);<br/></i><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E590</font></i>: 
 <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">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">FF</font> = <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
( <font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> &amp; <font color="Maroon">pf</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
( <font color="Maroon">pl</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E591</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LL</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E592</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E593</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</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="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/><i><font color="Green">E594</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">G1</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E595</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, <a class="ref" href="goboard1.html#T39">GOBOARD1:39</a></i>;<br/><div><i><font color="Green">E596</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><i><font color="Green">E597</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</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> )
 ;<br/><b>then reconsider </b><font color="Maroon">w3</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>, <font color="Maroon">w2</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w3</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><b>then </b><i><font color="Green">E600</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w2</font> )
 <b>by </b><i/>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/><i><font color="Green">E601</font></i>: 
<font color="Maroon">w2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font>
 ;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_5"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> )
 ;<br/><b>hence </b>1 = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E602</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">pf</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><i><font color="Green">E603</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b><i><font color="Green">E604</font></i>: 
( <font color="Maroon">h1</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">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">l1</font>,<font color="Maroon">K1</font> &amp; <font color="Maroon">l1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">l1</font>,<font color="Maroon">K1</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">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b><i><font color="Green">E605</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l1</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">w4</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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>  ;<br/><i><font color="Green">E607</font></i>: <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w4</font>
<b>by </b><i>, , <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w4</font>
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">mi</font>,<font color="Maroon">K1</font>
<b>by </b><i>, , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">mi</font>,<font color="Maroon">K1</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_6"/><b>then </b>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mi</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K1</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">mi</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">mi</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E20:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E608</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> holds <br/>for <font color="Olive">i1</font>, <font color="Olive">i2</font>, <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">j1</font>,<font color="Olive">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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">j1</font>,<font color="Olive">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="real_1.html#K5">-</a> <font color="Olive">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><b>set </b><font color="Maroon">hf</font> = <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><i><font color="Green">E609</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</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">G1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font> &amp; <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j1</font>,<font color="Maroon">j2</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> )
 ;<br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b><i><font color="Green">E610</font></i>: 
( <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K2</font> &amp; <font color="Maroon">ma</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">G1</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">K2</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">G1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b><i><font color="Green">E611</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><i><font color="Green">E612</font></i>: 
(  <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> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">LL</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><i><font color="Green">E614</font></i>: <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font>
<b>by </b><i>, , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font>
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">ma</font>,<font color="Maroon">K2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">ma</font>,<font color="Maroon">K2</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">G1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E10:1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_7"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ma</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K2</font> )
 <b>by </b><i><a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">ma</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">ma</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> 1
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E22:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</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">G1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</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">G1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="goboard4.html#E15:1_1_5_1_2_1_1_2"><i><font color="Green">E139</font></i></a>, <a class="ref" href="goboard1.html#T40">GOBOARD1:40</a></i>;<br/><a NAME="E23:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E615</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><div><i><font color="Green">E616</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1"/>
( <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E617</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><b>then </b>
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><b>then </b><i><font color="Green">E618</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E619</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pl</font> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/><i><font color="Green">E620</font></i>: 
<font color="Maroon">pf</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pf</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E13:1_1_5_1_2_1_1_2"><i><font color="Green">E138</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><i><font color="Green">E621</font></i>: 
<font color="Maroon">mi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">mi</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E622</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E623</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><i><font color="Green">E624</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E625</font></i>: 
<font color="Maroon">K1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b><i><font color="Green">E626</font></i>: 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K1</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
0 <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><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> 1,<font color="Maroon">K1</font>
<b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E45:1_1"><i><font color="Green">E112</font></i></a>, <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font>
<b>by </b><i>, , , , <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_8_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E627</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_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>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/><a NAME="E26:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
0 <a href="nat_1.html#K1">+</a> 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">h1</font></span>)</span> <a href="nat_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>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E27:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E628</font></i>: 
0 <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">F</font></span>)</span> <a href="nat_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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><div><i><font color="Green">E629</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1"/>
( <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> or <font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E630</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><b>then </b>
<font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><b>then </b><i><font color="Green">E631</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E632</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">pl</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">pl</font>
 ;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/><i><font color="Green">E633</font></i>: 
(  <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> &amp; <font color="Maroon">pl</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">pf</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E3:1_1_5_1_2_1_1_2"><i><font color="Green">E129</font></i></a>, <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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">h1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font>
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pl</font>
<b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>
;<br/><b>hence </b><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><i><font color="Green">E634</font></i>: 
<font color="Maroon">ma</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
<font color="Maroon">ma</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b>
<font color="Maroon">ma</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b>
<font color="Maroon">ma</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">G1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E635</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E636</font></i>: 
<font color="Maroon">l2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">l2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E637</font></i>: 
<font color="Maroon">K2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Ll</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><b>then </b><i><font color="Green">E638</font></i>: 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ll</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K2</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/><i><font color="Green">E639</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <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> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l2</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l2</font></span>)</span>,<font color="Maroon">K2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">Ll</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font>
<b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/><b>hence </b><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_10_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><i><font color="Green">E640</font></i>:  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
;<br/><a NAME="E30:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E641</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"><b>proof </b></a><div class="add">

<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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E642</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E643</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon">gg</font> =  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font>;<br/>
<b>assume </b><a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E644</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<b>then </b><i><font color="Green">E645</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/><i><font color="Green">E646</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">K2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E647</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E648</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E649</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">K1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E44:1_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E650</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">pf</font>
 ;<br/><i><font color="Green">E651</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK1</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E652</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK1</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><i><font color="Green">E653</font></i>: 
( <font color="Maroon">Lmi</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lmi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K1</font> &amp; <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">mi</font> )
 <b>by </b><i>, , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
<font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">CK1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b><i><font color="Green">E654</font></i>: 
( <font color="Maroon">K1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font> )
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">pf</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">CK1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">mi</font>
 <b>by </b><i>, , , , , <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E655</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">i</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="E4:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E656</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E657</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">K2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y</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> <b>by </b><i><a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E658</font></i>: 
<font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">LL</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E12:1_1_5_1_2_1_1_2"><i><font color="Green">E135</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">LL</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><i><font color="Green">E659</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK2</font> = 
 <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">G1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G1</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E660</font></i>: 
<font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">CK2</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/><a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><i><font color="Green">E661</font></i>: 
( <font color="Maroon">Lma</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Lma</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">K2</font> &amp; <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ma</font> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
<font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">ma</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E14:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">CK2</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E32:1_1_5_1_2_1_1_2"><i><font color="Green">E157</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><a NAME="E15:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b><i><font color="Green">E662</font></i>: 
<font color="Maroon">K2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="ref" href="goboard1.html#T20">GOBOARD1:20</a></i>;<br/><a NAME="E16:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E17:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R1">=</a> <font color="Maroon">CK2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ma</font>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E39:1_1"><i><font color="Green">E105</font></i></a>, , , , <a class="ref" href="goboard1.html#T17">GOBOARD1:17</a></i>;<br/><a NAME="E18:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">LL</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="txt" href="goboard4.html#E33:1_1_5_1_2_1_1_2"><i><font color="Green">E159</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/><b>hence </b><a NAME="E19:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_12"/>
contradiction
 ;<br/>


</div><b>end;</b></div><i><font color="Green">E663</font></i>: <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="goboard4.html#E2:1_1_5_1_2_1_1_2"><i><font color="Green">E128</font></i></a>, , <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>

;<br/><a NAME="E32:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E664</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E34:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"><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="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</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="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><i><font color="Green">E665</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font>
 
<b>by </b><i>, , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E6:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> )
 
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</font></i></a>, <a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_14"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E35:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E666</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="goboard4.html#E24:1_1_5_1_2_1_1_2"><i><font color="Green">E148</font></i></a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E36:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><i><font color="Green">E667</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E37:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E668</font></i>: 
not <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E3:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E669</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</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="E5:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E670</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T4">PARTFUN2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="goboard4.html#E4:1_1_5_1_2_1_1_2"><i><font color="Green">E130</font></i></a>, , , <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/>
<a NAME="E7:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E671</font></i>: 
( <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">pf</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> )
 
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a></i>;<br/>
<a NAME="E8:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
( 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> <font color="Maroon">LL</font> )
 
<b>by </b><i>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><b>then </b><i><font color="Green">E672</font></i>: 
( <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">pf</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">LL</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E10:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><i><font color="Green">E673</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g2</font>
 
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E11:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
<font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , <a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a>, <a class="txt" href="goboard4.html#E29:1_1_5_1_2_1_1_2"><i><font color="Green">E153</font></i></a>, <a class="txt" href="goboard4.html#E30:1_1_5_1_2_1_1_2"><i><font color="Green">E154</font></i></a>, <a class="ref" href="finseq_4.html#T86">FINSEQ_4:86</a></i>;<br/>
<a NAME="E12:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/><b>then </b>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, , <a class="txt" href="goboard4.html#E38:1_1"><i><font color="Green">E104</font></i></a>, , <a class="txt" href="goboard4.html#E31:1_1_5_1_2_1_1_2"><i><font color="Green">E156</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E13:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2_15"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E38:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="goboard4.html#E22:1_1_5_1_2_1_1_2"><i><font color="Green">E146</font></i></a></i>;<br/><a NAME="E39:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E40:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E674</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E31:1_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="goboard4.html#E16:1_1_5_1_2_1_1_2"><i><font color="Green">E140</font></i></a>, , <a class="txt" href="goboard4.html#E20:1_1_5_1_2_1_1_2"><i><font color="Green">E144</font></i></a>, , <a class="ref" href="goboard1.html#T37">GOBOARD1:37</a>, <a class="ref" href="goboard1.html#T41">GOBOARD1:41</a></i>;<br/><a NAME="E41:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="goboard1.html#K4">DelCol</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#T26">GOBOARD1:26</a></i>;<br/><a NAME="E42:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/><b>then </b><i><font color="Green">E675</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , , , <a class="txt" href="goboard4.html#E19:1_1_5_1_2_1_1_2"><i><font color="Green">E143</font></i></a>, <a class="txt" href="goboard4.html#E27:1_1_5_1_2_1_1_2"><i><font color="Green">E151</font></i></a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>;<br/><a NAME="E44:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">h2</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">tt</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard4.html#E28:1_1_5_1_2_1_1_2"><i><font color="Green">E152</font></i></a></i>;<br/><b>hence </b><a NAME="E45:1_1_5_1_2_1_1_2_22_1_2_1_1_2_17_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="goboard4.html#E25:1_1_5_1_2_1_1_2"><i><font color="Green">E149</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="E63:1_1_5_1_2_1_1_2_22_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2_1_1_2_22_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<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><b>end;</b></div><b>hence </b><a NAME="E4:1_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E47:1_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/><i><font color="Green">E676</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<div><i><font color="Green">E677</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>now </b></a><div class="add"><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>let </b><font color="Maroon">G1</font> be   <a href="goboard1.html#NM1">Go-board</a>;<br/><b>let </b><font color="Maroon">g1</font> be   <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>let </b><font color="Maroon">g2</font> be   <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>assume </b><a NAME="E1:1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">g2</font> <a href="goboard1.html#R1">is_sequence_on</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">g1</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">g1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G1</font></span>)</span></span>)</span> &amp; <font color="Maroon">g2</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,1</span>)</span> &amp; <font color="Maroon">g2</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">g2</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">G1</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">G1</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:1"/>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E7:1"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>hence </b><a NAME="E8:1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/>


</div>
