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<span class="kw">let </span><font color="Maroon" title="c1">j</font>, <font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  not <font color="Olive" title="b1">G</font> is  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="goboard1.html#V3" title="GOBOARD1:attr.3">Y_equal-in-column</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="goboard1.html#V4" title="GOBOARD1:attr.4">Y_increasing-in-line</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Olive" title="b1">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">G</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Olive" title="b1">G</font>,<font color="Maroon" title="c1">j</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">G</font> is  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V3" title="GOBOARD1:attr.3">Y_equal-in-column</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V4" title="GOBOARD1:attr.4">Y_increasing-in-line</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font> &amp; <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font> implies  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 not <font color="Maroon" title="c3">G</font> is  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a> 
 <span class="kw">and </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V3" title="GOBOARD1:attr.3">Y_equal-in-column</a> 
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V4" title="GOBOARD1:attr.4">Y_increasing-in-line</a> 
 <span class="kw">and </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">j</font>
 <span class="kw">and </span><br/><a NAME="E5:25"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">and </span><br/><a NAME="E6:25"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font>
 <span class="kw">and </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)</span><br/>

<a NAME="E8:25"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:25"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>) or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E10:25"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E12:25"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A12</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th2</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:25_2_1"/>
( <font color="Maroon" title="c1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font> or <font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:25_2_1_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)</span><br/><div class="add"><a NAME="E2:25_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/><a NAME="E3:25_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E34">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:25_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E1:25_2_1_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="goboard5.html#T6" target="_self" title="GOBOARD5:th.6">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:25_2_1_2"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)</span><br/><div class="add"><a NAME="E2:25_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A15</font></span>: 
<font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:25_2_1_2"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/><a NAME="E4:25_2_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E13:25"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></span>;<br/><a NAME="E5:25_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2_1_2"><span class="lab"><font color="Green" title="E38">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:25_2_1_2"/>
<font color="Maroon" title="c1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E7:25_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E2:25_2_1_2"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E5:25_2_1_2"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="ref" href="goboard5.html#T4" target="_self" title="GOBOARD5:th.4">Th5</a></span>;<br/><a NAME="E8:25_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">j</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E3:25_2_1_2"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:25_2_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E1:25_2_1_2"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="goboard5.html#T5" target="_self" title="GOBOARD5:th.5">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:25"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> (<font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">j</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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