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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="matrix_3.html#K11">Path_product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E2:44"/><i><font color="Green">E41</font></i>: 
 <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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="E3:44"/><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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:44"><i><font color="Green">E41</font></i></a>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<a NAME="E4:44"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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="E5:44"/><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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:44"/><i><font color="Green">E45</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#K7">$$</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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44"><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">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E8:44"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:44_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:44_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:44_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_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="E3:44_1"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#E4:44"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:44_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:44"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></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#E8:44"><i><font color="Green">E46</font></i></a>);<br/></i>
<a NAME="E11:44"/><i><font color="Green">E48</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<a NAME="E1:44_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="txt" href="matrix_7.html#E10:44"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E2:44_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:44_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>14</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>
<b>thus </b><a NAME="E4:44_2_1"/>
<font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E5:44"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E3:44_2_1"><i><font color="Green">E51</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="E3:44_2"/>
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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="E1:44_2_2"/>
<font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></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#E5:44"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K2">"</a> ;<br/>
<a NAME="E3:44_2_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T65">FUNCT_1:65</a></i>;<br/>
<a NAME="E4:44_2_2"/>
<font color="Maroon">c<sub>14</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#E5:44"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:44_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_2_2"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44_2"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:44_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>, <a class="txt" href="matrix_7.html#E7:44"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E1:44_2"><i><font color="Green">E49</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:44"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:44_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:44_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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>14</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#E4:44"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E4:44_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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="44_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></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="E1:44_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_3_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:44_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_3_1"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:44_3"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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="E6:44_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:44"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E1:44_3"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:44_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E10:44"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:44"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E3:44_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K2">"</a>  is    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E4:44_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></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="44_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_4_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></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="44_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:44_4_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_4_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>
<a NAME="E4:44_4_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="txt" href="matrix_7.html#E3:44_4_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:44_4_1_1"/>
<font color="Maroon">c<sub>16</sub></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">c<sub>15</sub></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#E3:44_4_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:44_4_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></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">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:44_4_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>15</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:44_4_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:44_4_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></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#E1:44_4_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:44_4_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:44_4_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></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">c<sub>15</sub></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#E1:44_4_1"><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">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>11</sub></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">c<sub>15</sub></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#E10:44"><i><font color="Green">E47</font></i></a></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">c<sub>3</sub></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">c<sub>15</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E7:44"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E3:44_4"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="matrix_7.html#T36" target="_self">Th36</a>, <a class="txt" href="matrix_7.html#E1:44"><i><font color="Green">E40</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:44_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E14:44"/>
for <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></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">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:44_5"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:44_5"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:44_5_1"/>
<font color="Maroon">c<sub>14</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:44_5_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>14</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:44_5_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<i><font color="Green">E54</font></i>: <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finsub_1.html#K1">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>)
<b>by </b><i><a class="txt" href="matrix_7.html#E10:44"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 

;<br/>
<a NAME="E5:44_5_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></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="E1:44_5_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_5_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:44_5_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#E3:44_5_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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#E4:44"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E8:44_5_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>16</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:44"><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="E9:44_5_1"/>
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K2">"</a>  as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="matrix_7.html#E8:44_5_1"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E11:44_5_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></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">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_5_1_3"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></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">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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="44_5_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:44_5_1_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:44_5_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></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="E3:44_5_1_3_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></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="44_5_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_5_1_3_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 </span><b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_1"><i><font color="Green">E57</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:44_5_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_5_1_3_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E4:44_5_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E5:44_5_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:44_5_1_3_1_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><div class="add"><a NAME="E2:44_5_1_3_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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="E3:44_5_1_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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="E4:44_5_1_3_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:44_5_1_3_1_1_1_2"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44_5_1_3_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:44_5_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:44_5_1_3_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_5_1_3_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>
<a NAME="E4:44_5_1_3_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_2"><i><font color="Green">E57</font></i></a>, <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="44_5_1_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_5_1_3_2_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="matrix_7.html#E4:44_5_1_3_2"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_2_1_1_1"><b>case </b></a><a NAME="E1:44_5_1_3_2_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:44_5_1_3_2_1_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_2"><i><font color="Green">E57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_5_1_3_2_1_1_2"><b>case </b></a><a NAME="E1:44_5_1_3_2_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:44_5_1_3_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:44_5_1_3_2_1_1_2"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_3_2"><i><font color="Green">E57</font></i></a>, <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="E6:44_5_1_3_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>17</sub></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="E12:44_5_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>15</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></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="E1:44_5_1_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_5_1_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E4:44_5_1_4"/>
<font color="Maroon">c<sub>18</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E3:44_5_1_4"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrix_7.html#E5:44"><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">c<sub>18</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font>;<br/>
<a NAME="E14:44_5_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:44_5"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E15:44_5_1"/>
<font color="Maroon">c<sub>18</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="txt" href="matrix_7.html#E1:44_5"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="matrix_7.html#E5:44"><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="E17:44_5_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K2">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E14:44_5_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E18:44_5_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E19:44_5_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>17</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:44"><i><font color="Green">E44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_5_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_5_1_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:44_5_1_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K2">"</a>  &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E4:44_5_1_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></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>, <a class="txt" href="matrix_7.html#E3:44_5_1_5"><i><font color="Green">E61</font></i></a></i>;<br/><b>hence </b><a NAME="E5:44_5_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:44_5_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="matrix_7.html#E3:44_5_1_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#T65">FUNCT_1:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E21:44_5_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E19:44_5_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E22:44_5_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K2">"</a> </span>)</span> where B is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:44"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E11:44_5_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="matrix_7.html#E12:44_5_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_7.html#E18:44_5_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E23:44_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E10:44"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E24:44_5_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>14</sub></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">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:44_5_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="matrix_7.html#E22:44_5_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="matrix_7.html#T36" target="_self">Th36</a>, <a class="txt" href="matrix_7.html#E1:44"><i><font color="Green">E40</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E15:44"/><b>then </b><i><font color="Green">E51</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K15">FinOmega</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#K7">$$</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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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#E3:44"><i><font color="Green">E42</font></i></a>, <a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a>, <a class="txt" href="matrix_7.html#E11:44"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrix_7.html#E12:44"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_7.html#E13:44"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E16:44"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>2</sub></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#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>3</sub></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="E17:44"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_3.html#K12">Det</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E15:44"><i><font color="Green">E51</font></i></a>, <a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>;<br/>


</div>
