<?xml version="1.0"?>
<div class="add">

<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">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b4">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b5">p</font> where <font color="Olive" title="b5">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">j</font> </span>}</span>  holds <br/>for <font color="Olive" title="b5">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> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b4">P</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b5">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b5">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></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">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b3">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b4">p</font> where <font color="Olive" title="b4">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> </span>}</span>  holds <br/>for <font color="Olive" title="b4">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> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b3">P</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b4">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b4">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</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"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies for <font color="Olive" title="b1">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> </span>}</span>  holds <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="Maroon" title="c2">K</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">P</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <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">P</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> </span>}</span>  holds <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="Maroon" title="c2">K</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">P</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b2">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span><br/>

<a NAME="E3:30"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n9</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">P</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">n91</font> = <font color="Maroon" title="c5">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">P1</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E5:30"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">aa</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<a NAME="E6:30"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T30" title="MATRIX_2:th.30">MATRIX_2:30</a>, <a class="ref" href="matrix_2.html#D17" title="MATRIX_2:def.17">MATRIX_2:def 17</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">PP</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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="c10">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> </span>}</span>  implies 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> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">PP</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E7:30"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c10">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> </span>}</span> 
 ; <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> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">PP</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">Proj</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">PP</font>,<span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:30"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c11">Proj</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E10:30"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
for <font color="Olive" title="b1">q</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> holds <br/><font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="laplace.html#T20" target="_self" title="LAPLACE:th.20">Th20</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">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"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">PP</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">DM</font> =  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">PathM</font> =  <a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">PathDM</font> =  <a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">pm</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">D</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">ProjD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font> holds <br/>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b2">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E11:30"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
for <font color="Olive" title="b1">B9</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">b</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">B9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">b</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font>] &amp; not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B9</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c18">b</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font>] &amp; not <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B9</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font>]
 <span class="kw">and </span><br/><a NAME="E2:30_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
not <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B9</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">S<sub>1</sub></font>[<font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:30_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">Proj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c17">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">ProjB9</font> = <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c17">B9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c20">D</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">ProjD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>  st <font color="Maroon" title="c20">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c20">D</font> &amp; <font color="Maroon" title="c20">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font> holds <br/>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c21">ProjD</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>; <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="c20">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c21">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c20">D</font> &amp; <font color="Maroon" title="c20">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font> implies the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:30_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c20">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:30_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c21">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c20">D</font>
 <span class="kw">and </span><br/><a NAME="E9:30_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c20">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E10:30_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c17">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E11:30_1"/>
<font color="Maroon" title="c17">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E12:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<font color="Maroon" title="c17">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E13:30_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E14:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">PP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">Q1</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E16:30_1"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
<font color="Maroon" title="c22">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">and </span><br/><a NAME="E17:30_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<font color="Maroon" title="c22">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E28">A5</font></span></a></span>;<br/>
<a NAME="E18:30_1"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">Proj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">PP</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E19:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A22</font></span>: 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c11">Proj</font>,<font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">Q</font> = <font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E18:30_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E21:30_1"/><span class="lab"><font color="Green" title="E46">A23</font></span>: 
<font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">Proj</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E18:30_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">Q19</font> = <font color="Maroon" title="c22">Q1</font> as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ;<br/>
<a NAME="E23:30_1"/><span class="lab"><font color="Green" title="E47">A24</font></span>: 
 <a href="laplace.html#K4" title="LAPLACE:func.4">Rem</a> <font color="Maroon" title="c24">Q19</font>,<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E16:30_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E17:30_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a></span>;<br/>
<a NAME="E24:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A25</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E17:30_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="ref" href="laplace.html#T22" target="_self" title="LAPLACE:th.22">Th22</a></span>;<br/>
<a NAME="E25:30_1"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
not <font color="Maroon" title="c23">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">ProjB9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c23">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">ProjB9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">Proj</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B9</font> &amp; <font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:30_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E18:30_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</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/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><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="E1:30_1_2"/>
( <font color="Maroon" title="c17">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c17">B9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1"/><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c17">B9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:30_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A28</font></span>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>;<br/><a NAME="E3:30_1_2_1"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Proj</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E19:30_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E21:30_1"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="setwiseo.html#T26" title="SETWISEO:th.26">SETWISEO:26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:30_1_2_1"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E16:30_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E17:30_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E23:30_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="txt" href="#E2:30_1_2_1"><span class="lab"><font color="Green" title="E51">A28</font></span></a>, <a class="ref" href="laplace.html#T22" target="_self" title="LAPLACE:th.22">Th22</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_2"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c17">B9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:30_1_2_2"/>
<font color="Maroon" title="c21">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">ProjB9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c23">Q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E8:30_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E19:30_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c21">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c19">ProjB9</font>,<span class="p3">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p4">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p4">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">Q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E12:30_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="txt" href="#E18:30_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="txt" href="#E25:30_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="txt" href="#E1:30_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c19">ProjB9</font>,<span class="p3">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p4">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p4">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p5">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p4">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c23">Q</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></span>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B9</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E16:30_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E10:30_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E24:30_1"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c20">D</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E7:30_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E1:30_1_2_2"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E12:30"/><span class="lab"><font color="Green" title="E32">A30</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>; <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">ProjD</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>  st <font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Olive" title="b1">ProjD</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font> holds <br/>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Olive" title="b1">ProjD</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c18">ProjB</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>; <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="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c18">ProjB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font> implies the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c18">ProjB</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c17">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:30_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c18">ProjB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c17">B</font>
 <span class="kw">and </span><br/><a NAME="E3:30_2"/>
<font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">PP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c18">ProjB</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E4:30_2"/>
<font color="Maroon" title="c18">ProjB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E2:30_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></span>;<br/>
<a NAME="E5:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A33</font></span>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c18">ProjB</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T10" title="FVSUM_1:th.10">FVSUM_1:10</a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></span>;<br/>
<a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
 <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T9" title="FVSUM_1:th.9">FVSUM_1:9</a></span>;<br/>
<a NAME="E7:30_2"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="ref" href="fvsum_1.html#T10" title="FVSUM_1:th.10">FVSUM_1:10</a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:30_2"/>
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c17">B</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c18">ProjB</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <span class="p3">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</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/>


</div><span class="kw">end;</span></div>
<a NAME="E13:30"/><span class="lab"><font color="Green" title="E33">A35</font></span>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E31">A8</font></span></a>);<br/></span>
<a NAME="E14:30"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">Proj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">PP</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:30"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">Proj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E16:30"/><span class="kw">then </span>
<font color="Maroon" title="c11">Proj</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c10">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E14:30"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/>
<span class="kw">hence </span>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c10">PP</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c12">M</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> <span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30"><span class="lab"><font color="Green" title="E33">A35</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K6" title="LAPLACE:func.6">Cofactor</a> <font color="Maroon" title="c12">M</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
;<br/> <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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