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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">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> ;<br/>





<b>assume </b><a NAME="E1:98"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c4">A</font> is <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<a NAME="E2:98"/><i><font color="Green" title="E66">B1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<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="E3:98"/><i><font color="Green" title="E67">B2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>thus </b><a NAME="E4:98"/>
( <font color="Maroon" title="c2">x</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>thus </b><font color="Maroon" title="c2">x</font> = 
<font color="Maroon" title="c2">x</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr2.html#T88" target="_self" title="MATRIXR2:th.88">AA3924a</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c4">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">AA444</a>, <a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr2.html#E1:98_1"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr2.html#T57" target="_self" title="MATRIXR2:th.57">AA516</a>, <a class="txt" href="matrixr2.html#E2:98"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrixr2.html#E3:98"><i><font color="Green" title="E67">B2</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:98"/><i><font color="Green" title="E68">A10</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span>
 ;<br/>

<b>thus </b><font color="Maroon" title="c3">y</font> = 
<font color="Maroon" title="c3">y</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr2.html#T88" target="_self" title="MATRIXR2:th.88">AA3924a</a></i>
<br/>.= 
<font color="Maroon" title="c3">y</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c4">A</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">AA444</a></i>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c4">A</font>
<b>by </b><i><a class="txt" href="matrixr2.html#E5:98"><i><font color="Green" title="E68">A10</font></i></a>, <a class="txt" href="matrixr2.html#E1:98"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr2.html#T57" target="_self" title="MATRIXR2:th.57">AA516</a>, <a class="txt" href="matrixr2.html#E2:98"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrixr2.html#E3:98"><i><font color="Green" title="E67">B2</font></i></a></i>
;<br/>


</div>
