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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/>  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b4">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">K</font> st <br/>(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b4">P</font>) &amp; (  for <font color="Olive" title="b5">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b5">perm</font>,<font color="Olive" title="b2">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b3">B</font></span>)</span>,<font color="Olive" title="b5">perm</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> holds <br/> ex <font color="Olive" title="b3">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b3">P</font>) &amp; (  for <font color="Olive" title="b4">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b3">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b4">perm</font>,<font color="Olive" title="b1">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b2">B</font></span>)</span>,<font color="Olive" title="b4">perm</font>)</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> implies  ex <font color="Olive" title="b1">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b1">P</font>) &amp; (  for <font color="Olive" title="b2">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b2">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b2">perm</font>)</span>)</span> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:84"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Olive" title="b1">P</font>) &amp; (  for <font color="Olive" title="b2">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b2">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b2">perm</font>)</span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">P</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E2:84"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">KK</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">mm</font> =  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<span class="kw">set </span><font color="Maroon" title="c9">AB</font> = <font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">X</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">F</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">SUM1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E4:84"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
 for <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>  ex <font color="Olive" title="b2">Path</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">Fj</font>, <font color="Olive" title="b4">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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> &amp; <font color="Olive" title="b3">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> holds <br/><font color="Olive" title="b2">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b4">j</font>,<font color="Olive" title="b3">Fj</font>) ) &amp; <font color="Maroon" title="c12">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b2">Path</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">F</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span></span>)</span>,<font color="Maroon" title="c12">SUM1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="matrix11.html#T60" target="_self" title="MATRIX11:th.60">Th60</a></span>;<br/>
<a NAME="E6:84"/><span class="lab"><font color="Green" title="E76">A5</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>) is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fraenkel.html#T6" title="FRAENKEL:th.6">FRAENKEL:6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">FP</font> = <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> 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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></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/>
<a NAME="E8:84"/><span class="lab"><font color="Green" title="E77">A6</font></span>: 
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c14">x</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E3:84_1"/>
<font color="Maroon" title="c15">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:84_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">P9</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E10:84"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</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> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T26" title="MATRIX_2:th.26">MATRIX_2:26</a>, <a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></span>;<br/>
<a NAME="E11:84"/><span class="lab"><font color="Green" title="E79">A8</font></span>: 
 <a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fraenkel.html#T6" title="FRAENKEL:th.6">FRAENKEL:6</a>, <a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E80">A9</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1"><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="E1:84_2_1"/>
( <font color="Maroon" title="c13">FP</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="c13">FP</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="84_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:84_2_1_1"/>
<font color="Maroon" title="c13">FP</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c14">P9</font>,<font color="Maroon" title="c12">SUM1</font>)</span><br/><div class="add"><a NAME="E2:84_2_1_1"/><span class="kw">then </span>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E3:84_2_1_1"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c14">P9</font>,<font color="Maroon" title="c12">SUM1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="txt" href="#E11:84"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="84_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:84_2_1_2"/><span class="lab"><font color="Green" title="E81">A10</font></span>: 
<font color="Maroon" title="c13">FP</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c14">P9</font>,<font color="Maroon" title="c12">SUM1</font>)</span><br/><div class="add"><a NAME="E2:84_2_1_2"/><span class="lab"><font color="Green" title="E82">A11</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> <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#T7" title="FVSUM_1:th.7">FVSUM_1:7</a></span>;<br/><a NAME="E3:84_2_1_2"/><span class="lab"><font color="Green" title="E83">A12</font></span>: 
<font color="Maroon" title="c12">SUM1</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c13">FP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SUM1</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c13">FP</font> or <font color="Maroon" title="c15">s</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_2_1_2_1"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">SUM1</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c13">FP</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">s</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:84_2_1_2_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">SUM1</font>
 <span class="kw">and </span><br/><a NAME="E4:84_2_1_2_1"/><span class="lab"><font color="Green" title="E84">A13</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FP</font>
 <span class="kw">and </span><br/><a NAME="E5:84_2_1_2_1"/><span class="lab"><font color="Green" title="E85">A14</font></span>: 
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c16">x</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_2_1_2_1"><span class="lab"><font color="Green" title="E84">A13</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/>
<a NAME="E7:84_2_1_2_1"/>
 not <font color="Maroon" title="c17">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84_2_1_2_1"><span class="lab"><font color="Green" title="E84">A13</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:84_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A15</font></span>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c17">f</font></span>)</span> <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="matrix11.html#T54" target="_self" title="MATRIX11:th.54">Th54</a></span>;<br/>
<a NAME="E9:84_2_1_2_1"/>
 ex <font color="Olive" title="b1">Path</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">Fj</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="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> &amp; <font color="Olive" title="b2">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Olive" title="b1">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">Fj</font>) ) &amp; <font color="Maroon" title="c12">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Olive" title="b1">Path</font></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"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c17">f</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E74">A3</font></span></a></span>;<br/>
<a NAME="E10:84_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">f</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="txt" href="#E8:84_2_1_2_1"><span class="lab"><font color="Green" title="E86">A15</font></span></a>, <a class="ref" href="vectsp_1.html#T6" title="VECTSP_1:th.6">VECTSP_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:84_2_1_2_1"/>
<font color="Maroon" title="c15">s</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"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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="#E5:84_2_1_2_1"><span class="lab"><font color="Green" title="E85">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:84_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">SUM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 <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="E5:84_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">SUM1</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c13">FP</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"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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="#E1:84_2_1_2"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="txt" href="#E3:84_2_1_2"><span class="lab"><font color="Green" title="E83">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E6:84_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A16</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="c13">FP</font>,<font color="Maroon" title="c12">SUM1</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="txt" href="#E2:84_2_1_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="ref" href="fvsum_1.html#T8" title="FVSUM_1:th.8">FVSUM_1:8</a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></span>;<br/><a NAME="E7:84_2_1_2"/><span class="lab"><font color="Green" title="E85">A17</font></span>: 
<font color="Maroon" title="c13">FP</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E8:84_2_1_2"/><span class="lab"><font color="Green" title="E86">A18</font></span>: 
<font color="Maroon" title="c13">FP</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/><a NAME="E9:84_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)</span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/><span class="kw">hence </span> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> = 
<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="c14">P9</font>,<font color="Maroon" title="c12">SUM1</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="txt" href="#E11:84"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="txt" href="#E6:84_2_1_2"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E7:84_2_1_2"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="txt" href="#E8:84_2_1_2"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="fvsum_1.html#T8" title="FVSUM_1:th.8">FVSUM_1:8</a>, <a class="ref" href="setwop_2.html#T4" title="SETWOP_2:th.4">SETWOP_2:4</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="c14">P9</font>,<font color="Maroon" title="c12">SUM1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_1:4</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E13:84"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">SUM1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>)
 
<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="E14:84"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A19</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">SUM1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E77">A6</font></span></a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a></span>;<br/>
<a NAME="E15:84"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">SUM1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<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>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">SP</font> = <font color="Maroon" title="c12">SUM1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E81">A19</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c15">SP</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">SP</font>) &amp; (  for <font color="Olive" title="b1">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c15">SP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b1">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b1">perm</font>)</span>)</span> ) )</span><br/>

<a NAME="E17:84"/><span class="lab"><font color="Green" title="E82">A20</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></span>;<br/>
<a NAME="E18:84"/>
<font color="Maroon" title="c15">SP</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<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>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">SP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E81">A19</font></span></a>, <a class="ref" href="relat_1.html#T52" title="RELAT_1:th.52">RELAT_1:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:84"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="matrix_2.html#K15" title="MATRIX_2:func.15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">SP</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:84"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="#E17:84"><span class="lab"><font color="Green" title="E82">A20</font></span></a>, <a class="txt" href="#E10:84"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="ref" href="setwop_2.html#T5" title="SETWOP_2:th.5">SETWOP_2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c15">SP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b1">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Olive" title="b1">perm</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">perm</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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">SP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c16">perm</font>)</span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c17">Perm</font> = <font color="Maroon" title="c16">perm</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E21:84"/>
<font color="Maroon" title="c12">SUM1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">Perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">SP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">Perm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:84"><span class="lab"><font color="Green" title="E81">A19</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">Path</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E23:84"/><span class="lab"><font color="Green" title="E83">A21</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E24:84"/><span class="lab"><font color="Green" title="E84">A22</font></span>: 
 for <font color="Olive" title="b1">Fj</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="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> &amp; <font color="Olive" title="b1">Fj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c18">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">Fj</font>)
 <span class="kw">and </span><br/><a NAME="E25:84"/><span class="lab"><font color="Green" title="E85">A23</font></span>: 
<font color="Maroon" title="c15">SP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">Perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <font color="Maroon" title="c18">Path</font></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"><font color="Maroon" title="c4">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c17">Perm</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E74">A3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">PM</font> =  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>);<br/>
<a NAME="E26:84"/><span class="lab"><font color="Green" title="E86">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:84_3"/><span class="lab"><font color="Green" title="E87">A25</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">Perm</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c20">i</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Path</font> implies <font color="Maroon" title="c18">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:84_3"/><span class="lab"><font color="Green" title="E88">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font>
 <span class="kw">and </span><br/><a NAME="E3:84_3"/><span class="lab"><font color="Green" title="E89">A27</font></span>: 
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">Path</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span><br/><a NAME="E4:84_3"/><span class="lab"><font color="Green" title="E90">A28</font></span>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:84_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A29</font></span>: 
<font color="Maroon" title="c20">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">by</span> <span class="lab"><a class="txt" href="#E23:84"><span class="lab"><font color="Green" title="E83">A21</font></span></a>, <a class="txt" href="#E2:84_3"><span class="lab"><font color="Green" title="E88">A26</font></span></a>, <a class="txt" href="#E3:84_3"><span class="lab"><font color="Green" title="E89">A27</font></span></a></span>;<br/><a NAME="E6:84_3"/>
<font color="Maroon" title="c20">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">by</span> <span class="lab"><a class="txt" href="#E23:84"><span class="lab"><font color="Green" title="E83">A21</font></span></a>, <a class="txt" href="#E2:84_3"><span class="lab"><font color="Green" title="E88">A26</font></span></a>, <a class="txt" href="#E3:84_3"><span class="lab"><font color="Green" title="E89">A27</font></span></a>, <a class="txt" href="#E4:84_3"><span class="lab"><font color="Green" title="E90">A28</font></span></a></span>;<br/><a NAME="E7:84_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A30</font></span>: 
<font color="Maroon" title="c17">Perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</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="c17">Perm</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_3"><span class="lab"><font color="Green" title="E87">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E8:84_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">Perm</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E9:84_3"/><span class="kw">then </span>
<font color="Maroon" title="c17">Perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">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">by</span> <span class="lab"><a class="txt" href="#E7:84_3"><span class="lab"><font color="Green" title="E92">A30</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">Pi</font> = <font color="Maroon" title="c17">Perm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> 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>  ;<br/><a NAME="E11:84_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:84"><span class="lab"><font color="Green" title="E86">A24</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E12:84_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">Pi</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84_3"><span class="lab"><font color="Green" title="E91">A29</font></span></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E13:84_3"/>
<font color="Maroon" title="c18">Path</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:84"><span class="lab"><font color="Green" title="E84">A22</font></span></a>, <a class="txt" href="#E5:84_3"><span class="lab"><font color="Green" title="E91">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E28:84"/><span class="kw">then </span>
<font color="Maroon" title="c18">Path</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:84"><span class="lab"><font color="Green" title="E83">A21</font></span></a>, <a class="txt" href="#E26:84"><span class="lab"><font color="Green" title="E86">A24</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:84"/>
<font color="Maroon" title="c15">SP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">K</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">perm</font>,<font color="Maroon" title="c3">A</font>)</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_2.html#K14" title="MATRIX_2:func.14">-</a> (<span class="p2">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c4">B</font></span>)</span>,<font color="Maroon" title="c16">perm</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:84"><span class="lab"><font color="Green" title="E85">A23</font></span></a>, <a class="ref" href="matrix11.html#T46" target="_self" title="MATRIX11:th.46">Th46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
