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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> &amp; <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> &amp; <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">M</font> = <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span>;<br/>
<a NAME="E1:10"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:10"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></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"> <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E3:10"/>
( <span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10"/>
<span class="p1"><a href="matrix_2.html#K5" title="MATRIX_2:func.5">&lt;*</a><span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span><a href="matrix_2.html#K5" title="MATRIX_2:func.5">*&gt;</a></span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="matrix_1.html#D5" title="MATRIX_1:def.5">MATRIX_1:def 5</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/>


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