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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c6">p</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> 2;<br/>











<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2
 ;<br/>

<a NAME="E2:16"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:16"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:16"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="matrix_9.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green" title="E17">A6</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E7:16"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="matrix_9.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<i><font color="Green" title="E19">A8</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1
<b>by </b><i><a class="txt" href="matrix_9.html#E4:16"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="matrix_9.html#E5:16"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,2
<b>by </b><i><a class="txt" href="matrix_9.html#E6:16"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="matrix_9.html#E7:16"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">d</font>
<b>by </b><i><a class="ref" href="matrix_2.html#T6" title="MATRIX_2:th.6">MATRIX_2:6</a></i>
;<br/>
<b>hence </b><a NAME="E10:16"/>
 <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="matrix_2.html#K6" title="MATRIX_2:func.6">][</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_9.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="matrix_9.html#E8:16"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>


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