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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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">A</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">f</font> =  <a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="matrix_3.html#K11">Path_product</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span>;<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">Y</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">B</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">n</font></span>)</span>;<br/>
<a NAME="E2:43"/><i><font color="Green">E40</font></i>: 
 <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/>
<a NAME="E3:43"/><i><font color="Green">E42</font></i>: 
(  <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">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green">E43</font></i>: 
 <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/>
<a NAME="E5:43"/><b>then </b><i><font color="Green">E44</font></i>: 
 <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">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">G0</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E45</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</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_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> <a href="funct_2.html#K3">.</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> &amp; ( for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) &amp; ( 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">n</font> holds  <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">B'</font> being   <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">n</font></span>)</span>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a>  <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">n</font></span>)</span> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>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">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</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> <a href="finsub_1.html#K2">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 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"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E8:43"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x2</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 ;<br/>

<a NAME="E2:43_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> )
 ;<br/>
<b>thus </b><a NAME="E4:43_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:43_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x2</font> </span>}</span>  is   <a href="finsub_1.html#NM1">Finite_Subset</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T34">FINSUB_1:34</a>, <a class="txt" href="matrix_7.html#E4:43"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">x2</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G1</font> being   <a href="funct_2.html#NM1">Function</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">n</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font><b> such that </b><br/><a NAME="E10:43"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x5</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x5</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x5</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="matrix_7.html#E8:43"><i><font color="Green">E46</font></i></a>);<br/></i>
<a NAME="E11:43"/><i><font color="Green">E53</font></i>: 
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_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">G1</font> <a href="funct_2.html#K3">.</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> <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_2"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <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">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_2_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:43_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43_2"/>
 <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">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q2</font> = <font color="Maroon">z</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a>, <a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">q3</font> = <font color="Maroon">q2</font> <a href="funct_2.html#K2">"</a> ;<br/>
<a NAME="E3:43_2_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T65">FUNCT_1:65</a></i>;<br/>
<a NAME="E4:43_2_2"/>
<font color="Maroon">q2</font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a>, <a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:43_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:43_2"/>
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_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</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>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="txt" href="matrix_7.html#E7:43"><i><font color="Green">E45</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:43"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>  st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>;<br/>

<b>assume </b><a NAME="E1:43_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:43_3"/>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T4">SUBSET_1:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B3</font> =  <a href="xboole_0.html#K1">{}</a>  as    <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">n</font></span>)</span> <b>by </b><i><a class="ref" href="finsub_1.html#T34">FINSUB_1:34</a>, <a class="txt" href="matrix_7.html#E4:43"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E4:43_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_3_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> )
 ;<br/>
<b>thus </b><a NAME="E4:43_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:43_3"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E6:43_3"/><b>then </b>
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B3</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:43"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:43_3"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:43"/><i><font color="Green">E62</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">n</font> holds  <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><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">n</font>;<br/>

<b>reconsider </b><font color="Maroon">B4</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> as    <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">n</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E3:43_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q</font> <a href="funct_2.html#K2">"</a>  is    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E4:43_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_4_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_4_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> )
 ;<br/>
<a NAME="E4:43_4_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="matrix_7.html#E11:43"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:43_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="matrix_7.html#E11:43"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:43_4_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_4_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:43_4_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:43_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="matrix_7.html#E11:43"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_4_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B4</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E10:43"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B4</font> = 
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E7:43"><i><font color="Green">E45</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:43"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_7.html#E2:43"><i><font color="Green">E40</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:43_4"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:43"/>
for <font color="Olive">B'</font> being   <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">n</font></span>)</span>  st <font color="Olive">B'</font> <a href="tarski.html#R1">c=</a>  <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">n</font></span>)</span> &amp; <font color="Olive">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>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">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</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> <a href="finsub_1.html#K2">\</a> <font color="Olive">B'</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 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"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B'</font> be    <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">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:43_5"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">B'</font> <a href="tarski.html#R1">c=</a>  <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">n</font></span>)</span> &amp; <font color="Maroon">B'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:43_5"/>
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">n</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</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> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B'</font> holds <br/><font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 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"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_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">n</font>;<br/>

<b>assume </b><a NAME="E1:43_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B'</font>
 ;<br/>

<a NAME="E2:43_5_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:43_5_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E4:43_5_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">G1</font> <a href="funct_2.html#K3">.</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> <a href="hidden.html#R1">=</a> <font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:43_5_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_5_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_5_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:43_5_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> </span>}</span>  as    <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">n</font></span>)</span> <b>by </b><i><a class="ref" href="finsub_1.html#T34">FINSUB_1:34</a>, <a class="txt" href="matrix_7.html#E4:43"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">px</font> = <font color="Maroon">x</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="txt" href="matrix_7.html#E11:43"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E8:43_5_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">px</font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E9:43_5_1"/>
 <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">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">px</font> <a href="funct_2.html#K2">"</a>  as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E11:43_5_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_5_1_2"/>
<font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_5_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:43_5_1_2_1"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:43_5_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_5_1_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:43_5_1_2_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_5_1_2_1_1_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 ;<br/><a NAME="E4:43_5_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:43_5_1_2_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p2</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p2</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:43_5_1_2_1_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:43_5_1_2_1_1_1_2"/>
<font color="Maroon">px</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:43_5_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">px</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:43_5_1_2_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43_5_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:43_5_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_5_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_5_1_2_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>
<a NAME="E4:43_5_1_2_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_5_1_2_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> or <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_2_1_1_1"><b>case </b></a><a NAME="E1:43_5_1_2_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_5_1_2_2_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_1_2_2_1_1_2"><b>case </b></a><a NAME="E1:43_5_1_2_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:43_5_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:43_5_1_2_2_1_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:43_5_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E12:43_5_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</a>  <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">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_5_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_5_1_3"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">p</font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 ;<br/>
<b>thus </b><a NAME="E4:43_5_1_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B'</font>;<br/>
<a NAME="E14:43_5_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:43_5_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">pd</font> = <font color="Maroon">d</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E17:43_5_1"/>
<font color="Maroon">pd</font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E18:43_5_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E19:43_5_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:43"><i><font color="Green">E44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_5_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/><b>then consider </b><font color="Maroon">pp</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:43_5_1_4"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pp</font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">pp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> )
 ;<br/><a NAME="E4:43_5_1_4"/>
<font color="Maroon">px</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">pp</font> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T65">FUNCT_1:65</a>, </i>;<br/><b>hence </b><a NAME="E5:43_5_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix_7.html#E10:43"><i><font color="Green">E52</font></i></a>, , <a class="ref" href="funct_1.html#T65">FUNCT_1:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:43_5_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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> <a href="finsub_1.html#K2">\</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E22:43_5_1"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">G0</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="funct_2.html#K2">"</a> </span>)</span> where <font color="Olive">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> : <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> </span>}</span>  <a href="hidden.html#R1">=</a> 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"><font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:43"><i><font color="Green">E45</font></i></a>, , , <a class="txt" href="matrix_7.html#E12:43"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E23:43_5_1"/>
<font color="Maroon">G0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E24:43_5_1"/>
<font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> 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"><font color="Maroon">G1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">B'</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="matrix_7.html#E1:43"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_7.html#E2:43"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E15:43"/><b>then </b><i><font color="Green">E115</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font> <a href="setwiseo.html#K7">$$</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_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:43"><i><font color="Green">E42</font></i></a>, <a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a>, <a class="ref" href="matrix_7.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E16:43"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 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_3.html#K11">Path_product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E17:43"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="matrix_3.html#K12">Det</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>;<br/>


</div>
