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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">i</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/>





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">G</font> is <a href="goboard1.html#V3">X_equal-in-line</a>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="goboard1.html#V6">X_increasing-in-column</a>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E43</font></i>: 
( 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="matrix_1.html#K1">width</a> <font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> )
 ;<br/>

<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="E5:30"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>,<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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:30"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/>
<a NAME="E8:30"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:30"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E11:30"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/>
<a NAME="E12:30"/><b>then </b>
<font color="Maroon">p</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">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</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">i</font>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</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">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="euclid.html#K21">`1</a>  ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="goboard5.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E13:30"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K1" target="_self">v_strip</a> <font color="Maroon">G</font>,<font color="Maroon">i</font>
 <b>by </b><i>, , , , </i>;<br/>


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