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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">M</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  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b2">M</font> <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"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></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">M</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  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Olive" title="b1">M</font> <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"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</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"> (  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> implies <font color="Maroon" title="c3">M</font> <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"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">C</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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">M</font> <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"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span><br/>

<a NAME="E2:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></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="ref" href="laplace.html#T30" target="_self" title="LAPLACE:th.30">Th30</a></span>;<br/>
<a NAME="E3:45"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c3">M</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="txt" href="#E1:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="laplace.html#T33" target="_self" title="LAPLACE:th.33">Th33</a></span>;<br/>
<a NAME="E4:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="matrix_6.html#R2" title="MATRIX_6:pred.2">is_reverse_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="matrix_6.html#D2" title="MATRIX_6:def.2">MATRIX_6:def 2</a></span>;<br/>
<a NAME="E5:45"/>
<font color="Maroon" title="c3">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="laplace.html#T34" target="_self" title="LAPLACE:th.34">Th34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:45"/>
<font color="Maroon" title="c3">M</font> <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"><span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="matrix_7.html#K1" title="MATRIX_7:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="laplace.html#K7" title="LAPLACE:func.7">Matrix_of_Cofactor</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="ref" href="matrix_6.html#D4" title="MATRIX_6:def.4">MATRIX_6:def 4</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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