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<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="matrix_1.html#NM4">Matrix</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> ) or <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>case </b></a><a NAME="E1:26_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1_1_1"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:26_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard5.html#D1">GOBOARD5:def 1</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_1"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:26_1_1_1_2"/><b>then </b>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 <b>by </b><i>, <a class="ref" href="goboard5.html#D1">GOBOARD5:def 1</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1_2"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:26_1_1_1"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>case </b></a><a NAME="E1:26_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard5.html#D1">GOBOARD5:def 1</a></i>;<br/><a NAME="E3:26_1_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1_2_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 ;<br/>

<a NAME="E2:26_1_1_2_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:26_1_1_2_1_1"/><b>then </b>
 ex <font color="Olive">s1</font>, <font color="Olive">r1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:26_1_1_2_1_1"/><i><font color="Green">E62</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">r1</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> )
 ;<br/>
<b>consider </b><font color="Maroon">s2</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:26_1_1_2_1_1"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">s2</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E8:26_1_1_2_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:26_1_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:26_1_1_2_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:26_1_1_2_1_2"/><b>then </b>
 ex <font color="Olive">s</font>, <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:26_1_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:26_1_1_2_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:26_1_1_2_1_3"/><b>then </b>
 ex <font color="Olive">s</font>, <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 
;<br/>
<b>hence </b><a NAME="E3:26_1_1_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:26_1_1_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1_1_2_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  <a href="xboole_0.html#K3">/\</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s2</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s2</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">P1</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">P2</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s1</font>,<font color="Olive">r1</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><a NAME="E6:26_1_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><a NAME="E7:26_1_1_2"/>
<font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:26_1_1_2"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , , <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_3"><b>case </b></a><a NAME="E1:26_1_1_3"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_3"/><b>then </b>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">r</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="goboard5.html#D1">GOBOARD5:def 1</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_3"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26"/>
 <a href="goboard5.html#K1">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">j</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


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