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





<b>assume </b><a NAME="E1:69"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font> ) )
 ;<br/>

<b>set </b><font color="Maroon">F</font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">K</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">M</font>;<br/>
<a NAME="E2:69"/>
 <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="E3:69"/><b>then </b><i><font color="Green">E34</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>reconsider </b><font color="Maroon">P1</font> =  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> 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="matrix_9.html#T4" target="_self">Th4</a></i>;<br/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">M</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> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
<b>by </b><i><a class="ref" href="matrix_9.html#T3" target="_self">Th3</a>, <a class="txt" href="matrix_9.html#E6"><i><font color="Green">Lemma53</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:69"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">K</font>
 ;<br/>


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