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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>, <font color="Maroon" title="c6">e</font>, <font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font>, <font color="Maroon" title="c10">i</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="c11">M</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of 3,<font color="Maroon" title="c1">K</font>;<br/>





<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">f</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c12">p</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 3;<br/>

<b>assume </b><a NAME="E2:32"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 ;<br/>

<a NAME="E3:32"/><i><font color="Green" title="E25">A5</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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E4:32"/><b>then </b><i><font color="Green" title="E26">A6</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="c12">p</font>,<font color="Maroon" title="c11">M</font></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> 3
 
<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="E5:32"/><b>then </b><i><font color="Green" title="E27">A7</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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span>
 
;<br/>
<a NAME="E6:32"/><i><font color="Green" title="E28">A8</font></i>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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#E2:32"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E7:32"/><i><font color="Green" title="E29">A9</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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E4:32"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green" title="E30">A10</font></i>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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#E2:32"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E9:32"/><i><font color="Green" title="E31">A11</font></i>: 
3 <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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E4:32"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<a NAME="E10:32"/><i><font color="Green" title="E32">A12</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 
<b>by </b><i><a class="txt" href="matrix_9.html#E2:32"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E11:32"/><i><font color="Green" title="E33">A13</font></i>: 
<font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E12:32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E14:32"/><i><font color="Green" title="E34">A14</font></i>: 
( <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c11">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E15:32"/>
<font color="Maroon" title="c13">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E11:32"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="matrix_9.html#E14:32"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E16:32"/><b>then </b><i><font color="Green" title="E35">A15</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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E5:32"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="matrix_9.html#E6:32"><i><font color="Green" title="E28">A8</font></i></a>, <a class="txt" href="matrix_9.html#E14:32"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E17:32"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c7">f</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E18:32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E20:32"/><i><font color="Green" title="E37">A17</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 &amp; <font color="Maroon" title="c11">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 2,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E21:32"/>
<font color="Maroon" title="c14">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E17:32"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="matrix_9.html#E20:32"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E22:32"/><b>then </b><i><font color="Green" title="E38">A18</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="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E7:32"><i><font color="Green" title="E29">A9</font></i></a>, <a class="txt" href="matrix_9.html#E8:32"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="matrix_9.html#E20:32"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E23:32"/><i><font color="Green" title="E39">A19</font></i>: 
<font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E24:32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">z</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font><b> such that </b><br/><a NAME="E26:32"/><i><font color="Green" title="E40">A20</font></i>: 
( <font color="Maroon" title="c15">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 &amp; <font color="Maroon" title="c11">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 3,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/>
<a NAME="E27:32"/>
<font color="Maroon" title="c15">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E23:32"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="matrix_9.html#E26:32"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E28:32"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E9:32"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="matrix_9.html#E10:32"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="matrix_9.html#E26:32"><i><font color="Green" title="E40">A20</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a></i>;<br/>
<b>hence </b><a NAME="E29:32"/>
 <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> <font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="matrix_9.html#E3:32"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="matrix_9.html#E16:32"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="matrix_9.html#E22:32"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>


</div>
