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<span class="kw">let </span><font color="Maroon" title="c1">a11</font>, <font color="Maroon" title="c2">a12</font>, <font color="Maroon" title="c3">a13</font>, <font color="Maroon" title="c4">a21</font>, <font color="Maroon" title="c5">a22</font>, <font color="Maroon" title="c6">a23</font>, <font color="Maroon" title="c7">a31</font>, <font color="Maroon" title="c8">a32</font>, <font color="Maroon" title="c9">a33</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> holds <br/>(  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Olive" title="b1">A</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Olive" title="b1">A</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Olive" title="b1">A</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">A</font> be   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> implies (  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:4"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )</span><br/>

<a NAME="E2:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <font color="Maroon" title="c10">A</font> <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="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c10">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c10">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 
;<br/>
<a NAME="E3:4"/>
( 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3 &amp; 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3 &amp; 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:4"/>
(  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c1">a11</font>,<font color="Maroon" title="c2">a12</font>,<font color="Maroon" title="c3">a13</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">a21</font>,<font color="Maroon" title="c5">a22</font>,<font color="Maroon" title="c6">a23</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> &amp;  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c10">A</font>,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c7">a31</font>,<font color="Maroon" title="c8">a32</font>,<font color="Maroon" title="c9">a33</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="matrix_0.html#T52" onmouseover="rs('matrix_0/T52')" onmouseout="rh()">MATRIX_0:52</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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