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<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">p</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>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="E1:42"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<a NAME="E2:42"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">K</font> <b>by </b><i><a class="ref" href="matrix_7.html#T1" target="_self">Th1</a>, <a class="txt" href="matrix_7.html#E2:42"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green">E42</font></i>: 
<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">p</font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">A</font></span>)</span></span>)</span>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</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">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="matrix_7.html#K4" target="_self">@</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>
<br/>.= 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="matrix_7.html#E2:42"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">p</font>
<b>by </b><i>, <a class="txt" href="matrix_7.html#E2:42"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">K</font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">A</font></span>)</span></span>)</span>,<font color="Maroon">p</font>
<b>by </b><i>, <a class="txt" href="matrix_7.html#E2:42"><i><font color="Green">E40</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E6:42"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrix_7.html#K3" target="_self">"</a> </span>)</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">p</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:42"><i><font color="Green">E42</font></i></a></i>;<br/>


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