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

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>





<b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M1</font> = <font color="Maroon">M</font> as   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n1</font>,<font color="Maroon">K</font> ;<br/>
<b>assume </b><a NAME="E3:68"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">F</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font>;<br/>
<b>set </b><font color="Maroon">P1</font> =  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>
<a NAME="E5:68"/>
 <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T30">MATRIX_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">P1</font> =  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span>] <b>means </b>the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>;<br/>
<a NAME="E7:68"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font>;<br/>

<b>assume </b><a NAME="E1:68_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 ;<br/>

the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i>, <a class="ref" href="matrix_9.html#T3" target="_self">Th3</a></i>
;<br/>
<b>hence </b><a NAME="E3:68_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:68"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">x</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font><br/> for <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Olive">x</font></span><a href="setwiseo.html#K2">.}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font>;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n1</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:68_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>] )
 ;<br/>

the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="setwop_2.html#T4">SETWOP_2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_9.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E3:68_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font> <a href="finsub_1.html#K1">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">x</font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:68"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">P1</font>]
 
<b>from </b><i><a class="ref" href="matrix_9.html#S1" target="_self">MATRIX_9:sch 1</a>(<a class="ref" href="matrix_9.html#T4" target="_self">Th4</a>, <a class="ref" href="matrix_9.html#T5" target="_self">Th5</a>);<br/></i>
<b>hence </b><a NAME="E10:68"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" target="_self">PPath_product</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/>


</div>
