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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</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="Red">::  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#V2" title="GOBOARD1:attr.2">X_equal-in-line</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="goboard1.html#V5" title="GOBOARD1:attr.5">X_increasing-in-column</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <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="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">j</font> &amp; <font color="Maroon" title="c2">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="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="c1">i</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Olive" title="b1">G</font>,<font color="Maroon" title="c1">i</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="Red">::  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#V2" title="GOBOARD1:attr.2">X_equal-in-line</a>  &amp; <font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V5" title="GOBOARD1:attr.5">X_increasing-in-column</a>  &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c2">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> 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="c1">i</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E22">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="E2:24"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V2" title="GOBOARD1:attr.2">X_equal-in-line</a> 
 <span class="kw">and </span><br/><a NAME="E3:24"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c3">G</font> is  <a href="goboard1.html#V5" title="GOBOARD1:attr.5">X_increasing-in-column</a> 
 <span class="kw">and </span><br/><a NAME="E4:24"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 <span class="kw">and </span><br/><a NAME="E5:24"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c1">i</font> <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">and </span><br/><a NAME="E6:24"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <span class="kw">and </span><br/><a NAME="E7:24"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c2">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="Red">::  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="c1">i</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font></span><br/>

<a NAME="E8:24"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">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="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">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="E9:24"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c2">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="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">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="Red">:: 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="Red">::  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="c1">i</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:24"/><span class="lab"><font color="Green" title="E31">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="c1">i</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</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="E12:24"/><span class="lab"><font color="Green" title="E32">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#T57" title="EUCLID:th.57">EUCLID:57</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">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="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`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="c1">i</font>,1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="goboard5.html#T3" target="_self" title="GOBOARD5:th.3">Th3</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> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="goboard5.html#T3" target="_self" title="GOBOARD5:th.3">Th3</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_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="E1:24_2_1"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font> or <font color="Maroon" title="c1">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> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E26">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="24_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_1"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="Red">::  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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font></span><br/><div class="add"><a NAME="E2:24_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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E3:24_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> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font></span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E32">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:24_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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:24_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="goboard5.html#T10" target="_self" title="GOBOARD5:th.10">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="24_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_1_2"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon" title="c1">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="Red">::  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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font></span><br/><div class="add"><a NAME="E2:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c1">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="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:24_2_1_2"/><span class="lab"><font color="Green" title="E36">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="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E4:24_2_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a></span>;<br/><a NAME="E5:24_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E36">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="E6:24_2_1_2"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">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="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><a NAME="E7:24_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E2:24_2_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E5:24_2_1_2"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="goboard5.html#T4" target="_self" title="GOBOARD5:th.4">Th4</a></span>;<br/><a NAME="E8:24_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="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E3:24_2_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:24_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#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E7:24"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E1:24_2_1_2"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="goboard5.html#T9" target="_self" title="GOBOARD5:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E15:24"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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