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<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">P2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_2_2"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i1</font>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i2</font>,<span class="p1">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) holds <br/>( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i1</font>,<font color="Olive">j2</font> )
 <b>and </b><br/><a NAME="E2:29_2_2"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i1</font>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i2</font>,<span class="p1">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) holds <br/>( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Olive">i1</font>,<font color="Olive">j2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_2_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1"><b>suppose </b></a><a NAME="E1:29_2_2_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:29_2_2_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:29_2_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><font color="Maroon">P1</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i>, , , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_2"><b>suppose </b></a><a NAME="E1:29_2_2_1_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:29_2_2_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:29_2_2_1_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><font color="Maroon">P1</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i>, , , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_3"><b>suppose </b></a><a NAME="E1:29_2_2_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:29_2_2_1_3"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:29_2_2_1_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><font color="Maroon">P1</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i>, , , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_4"><b>suppose </b></a><a NAME="E1:29_2_2_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:29_2_2_1_4"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:29_2_2_1_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><font color="Maroon">P1</font> = 
 <a href="goboard5.html#K3">cell</a> <font color="Maroon">G</font>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i>, , , , </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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