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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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">A</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Olive" title="b2">A</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  holds <br/>( <font color="Olive" title="b2">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Olive" title="b1">A</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  holds <br/>( <font color="Olive" title="b1">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <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_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <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="c3">A</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  implies ( <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:37"/>
<font color="Maroon" title="c3">A</font> is  <a href="matrix_6.html#V1" title="MATRIX_6: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="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">B</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c4">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:37"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T19" target="_self" title="MATRIX14:th.19">Th19</a></span>;<br/>
<a NAME="E5:37"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T30" target="_self" title="MATRIX14:th.30">Th30</a></span>;<br/>
<a NAME="E6:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix_6.html#T10" title="MATRIX_6:th.10">MATRIX_6:10</a></span>;<br/>
<a NAME="E7:37"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T30" target="_self" title="MATRIX14:th.30">Th30</a></span>;<br/>
<a NAME="E8:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="matrix_6.html#T10" title="MATRIX_6:th.10">MATRIX_6:10</a></span>;<br/>
<a NAME="E9:37"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:37"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix14.html#T18" target="_self" title="MATRIX14:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:37"/>
( <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a> </span>)</span> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span>)</span> <a href="matrix_1.html#K5" title="MATRIX_1:func.5">@</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E8:37"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="matrix14.html#T18" target="_self" title="MATRIX14:th.18">Th18</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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