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

<b>set </b><font color="Maroon">f1</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">f2</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E1:55_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E2:55_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font>;<br/>
<a NAME="E3:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E44</font></i>: <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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/>.= 
 <a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E5:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E45</font></i>: <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
 <a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E7:55_1_1"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E59</font></i>: <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 3
<b>by </b><i><a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>
<br/>.= 
 <a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T27">FINSEQ_4:27</a></i>
;<br/>
<a NAME="E9:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E63</font></i>: <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
 <a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E11:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E72</font></i>: <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_4.html#T84">FINSEQ_4:84</a></i>
<br/>.= 
 <a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/>
<a NAME="E13:55_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
;<br/>
<a NAME="E14:55_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,1 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E15:55_1_1"/>
2 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3
 
;<br/>
<a NAME="E16:55_1_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,2 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E17:55_1_1"/><i><font color="Green">E120</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,3 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K22">SE-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E18:55_1_1"/>
4 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 5
 
;<br/>
<a NAME="E19:55_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,4 <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K19">SW-corner</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E20:55_1_1"/>
 <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font> is <a href="topreal1.html#V1">special</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D7">TOPREAL1:def 7</a><br/>

<b>assume </b><a NAME="E1:55_1_1_6"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:55_1_1_6"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:55_1_1_6"/><b>then </b><i><font color="Green">E122</font></i>: 
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:55_1_1_6"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E5:55_1_1_6"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_6_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="ref" href="sprect_1.html#T5" target="_self">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_6_1_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="euclid.html#K22">`2</a>  = 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="sprect_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_6_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="sprect_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_6_1_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span> <a href="euclid.html#K22">`2</a>  = 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="sprect_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_6_1_4"><b>suppose </b></a><a NAME="E1:55_1_1_6_1_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 4</span>)</span> <a href="euclid.html#K21">`1</a>  = 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><b>hence </b><a NAME="E3:55_1_1_6_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="sprect_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E21:55_1_1"/>
 <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font> is <a href="topreal1.html#V2">unfolded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D8">TOPREAL1:def 8</a><br/>

<b>assume </b><a NAME="E1:55_1_1_7"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:55_1_1_7"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E3:55_1_1_7"/><b>then </b><i><font color="Green">E128</font></i>: 
( 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>assume </b><a NAME="E4:55_1_1_7"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E5:55_1_1_7"/><b>then </b><i><font color="Green">E129</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2 or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2 )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_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="E1:55_1_1_7_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3 )
 </span><b>by </b><i><a class="ref" href="sprect_1.html#T5" target="_self">Th5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_7_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="txt" href="sprect_1.html#E6:55_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="sprect_1.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_7_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="sprect_1.html#T3" target="_self">Th3</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_7_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_7_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_7_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="sprect_1.html#T4" target="_self">Th4</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E22:55_1_1"/>
 <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font> is <a href="finseq_6.html#V1">circular</a>
 <b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/>

<b>thus </b><a NAME="E23:55_1_1"/>
 <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font> is <a href="goboard5.html#V1">s.c.c.</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:55_1_1_8"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:55_1_1_8"/><i><font color="Green">E131</font></i>: 
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</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"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> ) or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> )
 ;<br/>

<a NAME="E3:55_1_1_8"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:55_1_1_8"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 4 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="sprect_1.html#T5" target="_self">Th5</a>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/>
<a NAME="E5:55_1_1_8"/><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E6:55_1_1_8"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:55_1_1_8"/><i><font color="Green">E136</font></i>: 
( ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 2 implies <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 ) )
 
;<br/>
<a NAME="E8:55_1_1_8"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E138</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_8_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i>, <a class="ref" href="sprect_1.html#T6" target="_self">Th6</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_8_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_8_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="sprect_1.html#T8" target="_self">Th8</a>, <a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_8_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_8_1_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="sprect_1.html#T8" target="_self">Th8</a>, <a class="ref" href="cqc_the1.html#T4">CQC_THE1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_1_1_3"><b>case </b></a><a NAME="E1:55_1_1_8_1_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_8_1_1_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="sprect_1.html#T5" target="_self">Th5</a>, <a class="ref" href="sprect_1.html#T7" target="_self">Th7</a>, <a class="ref" href="sprect_1.html#T8" target="_self">Th8</a>, , <a class="ref" href="cqc_the1.html#T5">CQC_THE1:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_8_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_2_1"><b>suppose </b></a><a NAME="E1:55_1_1_8_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:55_1_1_8_2_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:55_1_1_8_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_2_2"><b>suppose </b></a><a NAME="E1:55_1_1_8_2_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 )
 ;<br/><div class="add"><a NAME="E2:55_1_1_8_2_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="sprect_1.html#E6:55_1_1"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="sprect_1.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E3:55_1_1_8_2_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_8_2_3"><b>suppose </b></a><a NAME="E1:55_1_1_8_2_3"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 ;<br/><div class="add"><a NAME="E2:55_1_1_8_2_3"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T3" target="_self">Th3</a>, <a class="ref" href="sprect_1.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E3:55_1_1_8_2_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>set </b><font color="Maroon">Xf1</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">Xf2</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<b>reconsider </b><font color="Maroon">Xf</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E25:55_1_1"/><i><font color="Green">E142</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E26:55_1_1"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Xf</font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E27:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E144</font></i>: <font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E29:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E145</font></i>: <font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E31:55_1_1"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E146</font></i>: <font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E33:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E147</font></i>: <font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E35:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E148</font></i>: <font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_14"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Xf</font>
 ;<br/><a NAME="E2:55_1_1_14"/><b>then </b><i><font color="Green">E149</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 <b>by </b><i><a class="ref" href="sprect_1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_14_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_14_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_14_1_1"/>
<font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="sprect_1.html#T6" target="_self">Th6</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_14_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_14_1_2"/>
<font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_14_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_14_1_3"/>
<font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1_4"><b>suppose </b></a><a NAME="E1:55_1_1_14_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_14_1_4"/>
<font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, <a class="ref" href="sprect_1.html#T7" target="_self">Th7</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_14_1_5"><b>suppose </b></a><a NAME="E1:55_1_1_14_1_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_14_1_5"/>
<font color="Maroon">Xf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i><a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="sprect_1.html#T8" target="_self">Th8</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E38:55_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
<font color="Maroon">Xf</font> <a href="hidden.html#R1">=</a>  <a href="goboard1.html#K2">X_axis</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="sprect_1.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard1.html#D3">GOBOARD1:def 3</a></i>;<br/>
<a NAME="E39:55_1_1"/><i><font color="Green">E151</font></i>: 
 <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E152</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T102">ENUMSET1:102</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>
<br/>.= 
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="seq_4.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>
;<br/>
<b>then </b><i><font color="Green">E153</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Xf</font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="seq_4.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T42">ENUMSET1:42</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E43:55_1_1"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Xf</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green">E155</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Xf</font></span>)</span>
<b>by </b><i>, <a class="txt" href="sprect_1.html#E8:55_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E45:55_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> is <a href="goboard1.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a><br/>

<a NAME="E1:55_1_1_19"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E2:55_1_1_19"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:55_1_1_19"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:55_1_1_19"/><b>then </b><i><font color="Green">E157</font></i>: 
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:55_1_1_19"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E6:55_1_1_19"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E10:55_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E7:55_1_1_19"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E46:55_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">Yf1</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">Yf2</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E47:55_1_1"/><i><font color="Green">E161</font></i>: 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Yf</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E49:55_1_1"/><i><font color="Green">E163</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E50:55_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Yf</font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E51:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E165</font></i>: <font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E53:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E166</font></i>: <font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E55:55_1_1"/>
3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></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">E167</font></i>: <font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> 3 = 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 3
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
;<br/>
<a NAME="E57:55_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E168</font></i>: <font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E59:55_1_1"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></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">E169</font></i>: <font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K1">+</a> 2</span>)</span> = 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_25"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_25"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Yf</font>
 ;<br/><a NAME="E2:55_1_1_25"/><b>then </b><i><font color="Green">E170</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_25_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_25_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_25_1_1"/>
<font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_25_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_25_1_2"/>
<font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="ref" href="sprect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_25_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_25_1_3"/>
<font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1_4"><b>suppose </b></a><a NAME="E1:55_1_1_25_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_25_1_4"/>
<font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_25_1_5"><b>suppose </b></a><a NAME="E1:55_1_1_25_1_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>hence </b><a NAME="E2:55_1_1_25_1_5"/>
<font color="Maroon">Yf</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K22">`2</a> 
 <b>by </b><i><a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E62:55_1_1"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">Yf</font> <a href="hidden.html#R1">=</a>  <a href="goboard1.html#K3">Y_axis</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="goboard1.html#D4">GOBOARD1:def 4</a></i>;<br/>
<i><font color="Green">E172</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> = 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="enumset1.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T148">FINSEQ_2:148</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T70">ENUMSET1:70</a></i>
;<br/>
<a NAME="E64:55_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<b>then </b><i><font color="Green">E173</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Yf</font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="sprect_1.html#E12:55_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="tarski.html#K2">}</a></span>

;<br/>
<a NAME="E66:55_1_1"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Yf</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T147">FINSEQ_2:147</a></i>;<br/>
<i><font color="Green">E175</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">Yf</font></span>)</span>
<b>by </b><i><a class="txt" href="sprect_1.html#E10:55_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sprect_1.html#T11" target="_self">Th11</a>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>
;<br/>
<a NAME="E68:55_1_1"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> is <a href="goboard1.html#V1">increasing</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_29"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a><br/>

<a NAME="E1:55_1_1_29"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E2:55_1_1_29"/><b>then </b><i><font color="Green">E176</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E3:55_1_1_29"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 ;<br/>

<a NAME="E4:55_1_1_29"/><b>then </b><i><font color="Green">E177</font></i>: 
( ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 ) &amp; ( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 ) )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>assume </b><a NAME="E5:55_1_1_29"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E6:55_1_1_29"/><b>then </b>
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="sprect_1.html#T14" target="_self">Th14</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E7:55_1_1_29"/>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E69:55_1_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="goboard2.html#K2">Incr</a> <span class="p1">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T12" target="_self">Th12</a>, , <a class="ref" href="goboard2.html#D2">GOBOARD2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<i><font color="Green">E179</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<i><font color="Green">E180</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
2
<b>by </b><i><a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E72:55_1_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">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">n</font>,<font color="Olive">m</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"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">n</font>,<font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_32"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_1_1_32"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</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"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_32"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="matrix_2.html#T3">MATRIX_2:3</a></i>;<br/>
<a NAME="E3:55_1_1_32"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span></span><a href="enumset1.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T25">MCART_1:25</a></i>;<br/>
<a NAME="E4:55_1_1_32"/><i><font color="Green">E182</font></i>: 
( <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_32_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_32_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_32_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_32_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:55_1_1_32_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, <a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_32_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_32_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:55_1_1_32_1_2"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, <a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_32_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_32_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:55_1_1_32_1_3"/><b>then </b><i><font color="Green">E185</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, <a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_32_1_4"><b>suppose </b></a><a NAME="E1:55_1_1_32_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">m</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:55_1_1_32_1_4"/><b>then </b><i><font color="Green">E186</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p4">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p5">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="sprect_1.html#T13" target="_self">Th13</a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, <a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then </b><i><font color="Green">E187</font></i>: <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="matrix_2.html#K6">][</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">D</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> = 
 <a href="goboard2.html#K1">GoB</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2">X_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="goboard2.html#K2">Incr</a> <span class="p2">(<span class="default"><a href="goboard1.html#K3">Y_axis</a> <span class="p3">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sprect_1.html#T14" target="_self">Th14</a>, <a class="ref" href="sprect_1.html#T15" target="_self">Th15</a>, <a class="ref" href="goboard2.html#D1">GOBOARD2:def 1</a></i>
<br/>.= 
 <a href="goboard2.html#K3">GoB</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="goboard2.html#D3">GOBOARD2:def 3</a></i>
;<br/>
<a NAME="E74:55_1_1"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E75:55_1_1"/><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 2,2
 
<b>by </b><i><a class="ref" href="sprect_1.html#T2" target="_self">Th2</a>, <a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E76:55_1_1"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 3 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<a NAME="E77:55_1_1"/><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T6">MATRIX_2:6</a></i>;<br/>
<b>thus </b><a NAME="E78:55_1_1"/>
 <a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font> is <a href="goboard5.html#V2">standard</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_34"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:55_1_1_34"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a>,<a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_34_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:55_1_1_34_1"/>
<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"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E2:55_1_1_34_1"/><b>then </b><i><font color="Green">E192</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 5 )
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_34_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i>, <a class="ref" href="cqc_the1.html#T6">CQC_THE1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_34_1_1_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:55_1_1_34_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:55_1_1_34_1_1_1"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2
 <b>by </b><i><a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_34_1_1_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:55_1_1_34_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:55_1_1_34_1_1_2"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 2,2
 <b>by </b><i><a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1_3"><b>suppose </b></a><a NAME="E1:55_1_1_34_1_1_3"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b>
2
;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:55_1_1_34_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:55_1_1_34_1_1_3"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 2,1
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1_4"><b>suppose </b></a><a NAME="E1:55_1_1_34_1_1_4"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
1
;<br/><b>thus </b><a NAME="E2:55_1_1_34_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:55_1_1_34_1_1_4"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_1_1_5"><b>suppose </b></a><a NAME="E1:55_1_1_34_1_1_5"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
2
;<br/><b>thus </b><a NAME="E2:55_1_1_34_1_1_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><b>thus </b><a NAME="E3:55_1_1_34_1_1_5"/>
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</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"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> 1,2
 <b>by </b><i>, <a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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

<b>assume </b><b>that </b><br/><a NAME="E2:55_1_1_34"/><i><font color="Green">E198</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"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 <b>and </b><br/><a NAME="E3:55_1_1_34"/><i><font color="Green">E199</font></i>: 
<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> <span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E4:55_1_1_34"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:55_1_1_34"/><i><font color="Green">E201</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> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:55_1_1_34"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E7:55_1_1_34"/><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span> <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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>

<a NAME="E8:55_1_1_34"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:55_1_1_34"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 4 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="sprect_1.html#E2:55_1_1"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:55_1_1_34"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 4
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_34_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="ref" href="sprect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="sprect_1.html#T24" target="_self">Th24</a>, <a class="ref" href="cqc_the1.html#T5">CQC_THE1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_2_1"><b>suppose </b></a><a NAME="E1:55_1_1_34_2_1"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:55_1_1_34_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:55_1_1_34_2_1"/><b>then </b><i><font color="Green">E207</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:55_1_1_34_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:55_1_1_34_2_1"/><b>then </b><i><font color="Green">E208</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:55_1_1_34_2_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_34_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_2_2"><b>suppose </b></a><a NAME="E1:55_1_1_34_2_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:55_1_1_34_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:55_1_1_34_2_2"/><b>then </b><i><font color="Green">E210</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="sprect_1.html#T18" target="_self">Th18</a>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:55_1_1_34_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:55_1_1_34_2_2"/><b>then </b><i><font color="Green">E211</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:55_1_1_34_2_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_34_2_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_2_3"><b>suppose </b></a><a NAME="E1:55_1_1_34_2_3"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:55_1_1_34_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:55_1_1_34_2_3"/><b>then </b><i><font color="Green">E213</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:55_1_1_34_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:55_1_1_34_2_3"/><b>then </b><i><font color="Green">E214</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:55_1_1_34_2_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_34_2_3"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_34_2_4"><b>suppose </b></a><a NAME="E1:55_1_1_34_2_4"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:55_1_1_34_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,1</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E3:55_1_1_34_2_4"/><b>then </b><i><font color="Green">E216</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E4:55_1_1_34_2_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,2</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="goboard2.html#K3">GoB</a> <span class="p2">(<span class="default"><a href="sprect_1.html#K1" target="_self">SpStSeq</a> <font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="matrix_2.html#T4">MATRIX_2:4</a></i>;<br/><a NAME="E5:55_1_1_34_2_4"/><b>then </b><i><font color="Green">E217</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="sprect_1.html#E4:55_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="sprect_1.html#T17" target="_self">Th17</a>, , , <a class="ref" href="sprect_1.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:55_1_1_34_2_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><b>hence </b><a NAME="E7:55_1_1_34_2_4"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , <a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
