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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 2,<font color="Maroon" title="c1">D</font> holds  <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2</span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 2,<font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2</span>)</span></span><br/>



<b>reconsider </b><font color="Maroon" title="c3">AA</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2</span>)</span> as   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 2,<font color="Maroon" title="c1">D</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></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> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c3">AA</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> )</span><br/><b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E21">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></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> <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">AA</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><br/><a NAME="E2:23_1"/>
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">A</font> <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> 2</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E3:23_1"/><b>then </b>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default">1,2</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="txt" href="#E1:23_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:23_1"/><b>then </b>
( ( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp; ( <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:23_1"/>
<font color="Maroon" title="c3">AA</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2</span>)</span> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2</span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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