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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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>  &amp; <font color="Olive" title="b2">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  holds <br/>( <font color="Olive" title="b1">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Olive" title="b2">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  &amp;  <a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Olive" title="b2">B</font></span>)</span> <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="Olive" title="b1">A</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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="Red">::  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>  &amp; <font color="Maroon" title="c3">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  implies ( <font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  &amp;  <a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <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> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:91"/><span class="lab"><font color="Green" title="E60">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E2:91"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
<font color="Maroon" title="c3">B</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>  &amp;  <a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <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> )</span><br/>

<span class="lab"><font color="Green" title="E62">A3</font></span>: <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="c3">B</font></span>)</span> <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="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p3">(<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="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T28" target="_self" title="MATRIXR2:th.28">Th28</a></span>
<br/>.= 
<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="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T28" target="_self" title="MATRIXR2:th.28">Th28</a></span>
<br/>.= 
<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="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p2">(<span class="default"><a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T71" target="_self" title="MATRIXR2:th.71">Th71</a></span>
<br/>.= 
 <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="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>
;<br/>
<span class="lab"><font color="Green" title="E63">A4</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</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="c3">B</font></span>)</span> <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> = 
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p3">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T28" target="_self" title="MATRIXR2:th.28">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p3">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T28" target="_self" title="MATRIXR2:th.28">Th28</a></span>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font></span>)</span> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91"><span class="lab"><font color="Green" title="E61">A2</font></span></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>
<br/>.= 
<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#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T28" target="_self" title="MATRIXR2:th.28">Th28</a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr2.html#T71" target="_self" title="MATRIXR2:th.71">Th71</a></span>
<br/>.= 
 <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="txt" href="#E1:91"><span class="lab"><font color="Green" title="E60">A1</font></span></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:91"/>
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font> is  <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="matrixr2.html#D5" target="_self" title="MATRIXR2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <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></span><br/>

<span class="kw">hence </span><a NAME="E6:91"/>
 <a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c3">B</font></span>)</span> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:91"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E63">A4</font></span></a>, <a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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