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

<a NAME="E1:41_7_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> where <font color="Olive" title="b1">q</font> is   <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> : ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> ) </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">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="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> where <font color="Olive" title="b1">q</font> is   <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> : ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> ) </span> } </span>  or <font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_7_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> where <font color="Olive" title="b1">q</font> is   <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> : ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</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="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E2:41_7_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being   <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> st <br/>( <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:41_7_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">B</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> where <font color="Olive" title="b1">q</font> is   <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> : ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> ) </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp;  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> &amp; <font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> ($1,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) ) );<br/>
<span class="kw">set </span><font color="Maroon" title="c3">Y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span> ;<br/>
<a NAME="E3:41_7_1"/><span class="lab"><font color="Green" title="E42">A121</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_7_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span><br/>

<a NAME="E2:41_7_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">j</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp;  ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> &amp; <font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">j</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) ) )
 
;<br/>
<a NAME="E3:41_7_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D4" title="MATRIX_1:def.4">MATRIX_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_7_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</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>
<a NAME="E4:41_7_1"/><span class="lab"><font color="Green" title="E43">A122</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></span>
<a NAME="E5:41_7_1"/>
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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T33" title="GOBOARD7:th.33">GOBOARD7:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:41_7_1"/><span class="lab"><font color="Green" title="E44">A123</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">and </span><br/><a NAME="E8:41_7_1"/><span class="lab"><font color="Green" title="E45">A124</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">j</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:41_7_1"/><span class="lab"><font color="Green" title="E46">A125</font></span>: 
<font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c5">j</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T8" target="_self" title="JORDAN5D:th.8">Th10</a></span>;<br/>
<a NAME="E10:41_7_1"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_7_1"><span class="lab"><font color="Green" title="E44">A123</font></span></a>, <a class="txt" href="#E8:41_7_1"><span class="lab"><font color="Green" title="E45">A124</font></span></a>, <a class="txt" href="#E9:41_7_1"><span class="lab"><font color="Green" title="E46">A125</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Y</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">j</font> where <font color="Olive" title="b1">j</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_7_1"><span class="lab"><font color="Green" title="E42">A121</font></span></a>, <a class="txt" href="#E4:41_7_1"><span class="lab"><font color="Green" title="E43">A122</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">i1</font> =  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> ;<br/>
<a NAME="E12:41_7_1"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E14:41_7_1"/><span class="lab"><font color="Green" title="E47">A126</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font>
 <span class="kw">and </span><br/><a NAME="E15:41_7_1"/><span class="lab"><font color="Green" title="E48">A127</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:41_7_1"/><span class="lab"><font color="Green" title="E49">A128</font></span>: 
 ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">g</font> &amp; <font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">j</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) )
 ;<br/>
<a NAME="E17:41_7_1"/><span class="lab"><font color="Green" title="E50">A129</font></span>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_7_1"><span class="lab"><font color="Green" title="E47">A126</font></span></a>, <a class="txt" href="#E15:41_7_1"><span class="lab"><font color="Green" title="E48">A127</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E18:41_7_1"/><span class="lab"><font color="Green" title="E51">A130</font></span>: 
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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_7_1"><span class="lab"><font color="Green" title="E48">A127</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E19:41_7_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_7_1"><span class="lab"><font color="Green" title="E47">A126</font></span></a>, <a class="txt" href="#E15:41_7_1"><span class="lab"><font color="Green" title="E48">A127</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E20:41_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A131</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <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"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></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="#E17:41_7_1"><span class="lab"><font color="Green" title="E50">A129</font></span></a>, <a class="txt" href="#E18:41_7_1"><span class="lab"><font color="Green" title="E51">A130</font></span></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></span>;<br/>
<a NAME="E21:41_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A132</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T40" target="_self" title="JORDAN5D:th.40">Th42</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:41_7_1"/><span class="lab"><font color="Green" title="E54">A133</font></span>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">and </span><br/><a NAME="E24:41_7_1"/><span class="lab"><font color="Green" title="E55">A134</font></span>: 
<font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">j</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:41_7_1"><span class="lab"><font color="Green" title="E49">A128</font></span></a></span>;<br/>
<a NAME="E25:41_7_1"/><span class="lab"><font color="Green" title="E56">A135</font></span>: 
<font color="Maroon" title="c10">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="c1">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:41_7_1"><span class="lab"><font color="Green" title="E54">A133</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E26:41_7_1"/><span class="lab"><font color="Green" title="E57">A136</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:41_7_1"><span class="lab"><font color="Green" title="E54">A133</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A137</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_7_1_3_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="E1:41_7_1_3_1"/>
( <font color="Maroon" title="c10">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="c1">g</font> or <font color="Maroon" title="c10">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="c1">g</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:41_7_1"><span class="lab"><font color="Green" title="E56">A135</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="41_7_1_3_1_1"><span class="kw">case </span></a><a NAME="E1:41_7_1_3_1_1"/>
<font color="Maroon" title="c10">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="c1">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span><br/><div class="add"><a NAME="E2:41_7_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">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="c1">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="E3:41_7_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">g</font>,<font color="Maroon" title="c10">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:41_7_1"><span class="lab"><font color="Green" title="E57">A136</font></span></a>, <a class="ref" href="topreal1.html#T21" title="TOPREAL1:th.21">TOPREAL1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_7_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_7_1"><span class="lab"><font color="Green" title="E47">A126</font></span></a>, <a class="txt" href="#E24:41_7_1"><span class="lab"><font color="Green" title="E55">A134</font></span></a>, <a class="ref" href="sppol_2.html#T17" title="SPPOL_2:th.17">SPPOL_2:17</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="41_7_1_3_1_2"><span class="kw">case </span></a><a NAME="E1:41_7_1_3_1_2"/>
<font color="Maroon" title="c10">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="c1">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span><br/><div class="add"><a NAME="E2:41_7_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c1">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="jordan5d.html#E53"><span class="lab"><font color="Green" title="E41">Lm9</font></span></a>, <a class="ref" href="jordan5d.html#T1" target="_self" title="JORDAN5D:th.1">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:41_7_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_7_1"><span class="lab"><font color="Green" title="E47">A126</font></span></a>, <a class="txt" href="#E24:41_7_1"><span class="lab"><font color="Green" title="E55">A134</font></span></a>, <a class="ref" href="sppol_2.html#T17" title="SPPOL_2:th.17">SPPOL_2:17</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>
<a NAME="E28:41_7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:41_7_1"><span class="lab"><font color="Green" title="E52">A131</font></span></a>, <a class="ref" href="jordan5d.html#T40" target="_self" title="JORDAN5D:th.40">Th42</a></span>;<br/>
<a NAME="E29:41_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A138</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span> where <font color="Olive" title="b1">q</font> is   <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> : ( <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:41_7_1"><span class="lab"><font color="Green" title="E58">A137</font></span></a></span>;<br/>
<a NAME="E30:41_7_1"/>
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> holds <br/><font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <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="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies <font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_7_1_4"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</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="c11">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<a NAME="E2:41_7_1_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being   <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> st <br/>( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:41_7_1_4"/>
<font color="Maroon" title="c11">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></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="jordan5d.html#E50"><span class="lab"><font color="Green" title="E38">Lm6</font></span></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>
<a NAME="E31:41_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A139</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></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="#E29:41_7_1"><span class="lab"><font color="Green" title="E59">A138</font></span></a>, <a class="ref" href="seq_4.html#T43" title="SEQ_4:th.43">SEQ_4:43</a></span>;<br/>
<a NAME="E32:41_7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  is    <a href="xxreal_2.html#M2" title="XXREAL_2:mode.2">LowerBound</a> of <font color="Maroon" title="c2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_7_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">r</font> be   <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xxreal_2.html#D2" title="XXREAL_2:def.2">XXREAL_2:def 2</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="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</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="c11">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_7_1_5"/>
<font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</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="c11">r</font></span><br/>

<a NAME="E2:41_7_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A140</font></span>: 
 ex <font color="Olive" title="b1">q1</font> being   <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> st <br/>( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font> )
 
;<br/>
<a NAME="E3:41_7_1_5"/>
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> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard7.html#T33" title="GOBOARD7:th.33">GOBOARD7:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:41_7_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</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="c11">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_7_1_5"><span class="lab"><font color="Green" title="E61">A140</font></span></a>, <a class="ref" href="jordan5d.html#T31" target="_self" title="JORDAN5D:th.31">Th33</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>
<a NAME="E33:41_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">B</font> is  <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">bounded_below</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D9" title="XXREAL_2:def.9">XXREAL_2:def 9</a></span>;<br/>
<a NAME="E34:41_7_1"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">B</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></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="#E29:41_7_1"><span class="lab"><font color="Green" title="E59">A138</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c6">Y</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c2">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:41_7_1"><span class="lab"><font color="Green" title="E60">A139</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
 <a href="pscomp_1.html#K1" title="PSCOMP_1:func.1">lower_bound</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">proj1</a> <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">|</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">N-most</a> <span class="p3">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan5d.html#T15" target="_self" title="JORDAN5D:th.15">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E36:41_7_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="goboard2.html#K2" title="GOBOARD2:func.2">GoB</a> <font color="Maroon" title="c1">g</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K20" title="PSCOMP_1:func.20">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">g</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_7_1"><span class="lab"><font color="Green" title="E47">A126</font></span></a>, <a class="txt" href="#E15:41_7_1"><span class="lab"><font color="Green" title="E48">A127</font></span></a>, <a class="txt" href="#E21:41_7_1"><span class="lab"><font color="Green" title="E53">A132</font></span></a>, <a class="ref" href="euclid.html#T53" title="EUCLID:th.53">EUCLID:53</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>
