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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c3">nt1</font>, <font color="Maroon" title="c4">nt2</font>, <font color="Maroon" title="c5">nt</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c6">M</font> be   <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c2">K</font>;<br/><b>let </b><font color="Maroon" title="c7">perm</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>













<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c3">nt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">nt2</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">perm</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">Perm</font> = <font color="Maroon" title="c7">perm</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E3:36"/>
 <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">nt1</font>,<font color="Maroon" title="c5">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt2</font>,<font color="Maroon" title="c5">nt</font></span>)</span> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c8">Perm</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E1:36"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="matrix13.html#T33" target="_self" title="MATRIX13:th.33">TT33</a></i>;<br/>
<b>hence </b><a NAME="E4:36"/>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c3">nt1</font>,<font color="Maroon" title="c5">nt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c4">nt2</font>,<font color="Maroon" title="c5">nt</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">perm</font>
 <b>by </b><i><a class="ref" href="matrix11.html#T46" title="MATRIX11:th.46">MATRIX11:46</a></i>;<br/>

<b>thus </b> <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c3">nt1</font></span>)</span> = 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c3">nt1</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixr2.html#T43" title="MATRIXR2:th.43">MATRIXR2:43</a></i>
<br/>.= 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c4">nt2</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c8">Perm</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix13.html#E1:36"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="matrix13.html#T34" target="_self" title="MATRIX13:th.34">TT34</a></i>
<br/>.= 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c4">nt2</font></span>)</span> <a href="matrix_7.html#K4" title="MATRIX_7:func.4">@</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">perm</font>
<b>by </b><i><a class="ref" href="matrix11.html#T46" title="MATRIX11:th.46">MATRIX11:46</a></i>
<br/>.= 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">M</font>,<font color="Maroon" title="c5">nt</font>,<font color="Maroon" title="c4">nt2</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">perm</font>
<b>by </b><i><a class="ref" href="matrixr2.html#T43" title="MATRIXR2:th.43">MATRIXR2:43</a></i>
;<br/>


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