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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">A</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  holds <br/> <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  implies  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:89"/>
<font color="Maroon" title="c2">A</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span><br/>

<a NAME="E2:89"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>;<br/>
<a NAME="E3:89"/><span class="kw">then </span>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p2">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T72" target="_self" title="MATRIXR2:th.72">Th72</a></span>;<br/>
<a NAME="E4:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p2">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T46" target="_self" title="MATRIXR2:th.46">Th46</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:89_1"/>
(  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_1"><span class="kw">suppose </span></a><a NAME="E1:89_1_1"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:89_1_1"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="89_1_2"><span class="kw">suppose </span></a><a NAME="E1:89_1_2"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:89_1_2"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89"><span class="lab"><font color="Green" title="E60">A1</font></span></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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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