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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">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="E1:31"/><i><font color="Green" title="E24">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#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> 3;<br/>

<b>assume </b><a NAME="E2:31"/><i><font color="Green" title="E25">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">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 ;<br/>

<a NAME="E3:31"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<b>by </b><i><a class="ref" href="matrix_1.html#T20" title="MATRIX_1:th.20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green" title="E28">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="E6:31"/><b>then </b><i><font color="Green" title="E29">A6</font></i>: 
 <a href="setwop_2.html#K3" title="SETWOP_2:func.3">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="setwop_2.html#K2" title="SETWOP_2: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="E7:31"/><i><font color="Green" title="E30">A7</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K3" title="SETWOP_2:func.3">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#E6:31"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E8:31"/><i><font color="Green" title="E31">A8</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> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E9:31"/><i><font color="Green" title="E32">A9</font></i>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K3" title="SETWOP_2:func.3">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#E6:31"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E10:31"/><i><font color="Green" title="E33">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="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E11:31"/><i><font color="Green" title="E34">A11</font></i>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwop_2.html#K3" title="SETWOP_2:func.3">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#E6:31"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E12:31"/><i><font color="Green" title="E35">A12</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> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E2:31"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E13:31"/><i><font color="Green" title="E36">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#E1:31"><i><font color="Green" title="E24">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="E14:31"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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="txt" href="matrix_9.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</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="E16:31"/><i><font color="Green" title="E37">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> 1,1 <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> 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="E17:31"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c13">r</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="c2">a</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E13:31"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="matrix_9.html#E16:31"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>
<a NAME="E18:31"/><i><font color="Green" title="E39">A16</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="c2">a</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E7:31"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="matrix_9.html#E8:31"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E17:31"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="matrix_9.html#E16:31"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>
<a NAME="E19:31"/><i><font color="Green" title="E40">A17</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#E1:31"><i><font color="Green" title="E24">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="E20:31"/>
<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="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a>, <a class="txt" href="matrix_9.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green" title="E27">A4</font></i></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="E22:31"/><i><font color="Green" title="E41">A18</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="E23:31"/><i><font color="Green" title="E42">A19</font></i>: 
<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="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E19:31"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="matrix_9.html#E22:31"><i><font color="Green" title="E41">A18</font></i></a></i>;<br/>
<a NAME="E24:31"/><i><font color="Green" title="E43">A20</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#E9:31"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="matrix_9.html#E10:31"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E23:31"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="matrix_9.html#E22:31"><i><font color="Green" title="E41">A18</font></i></a></i>;<br/>
<a NAME="E25:31"/><i><font color="Green" title="E44">A21</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#E1:31"><i><font color="Green" title="E24">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="E26:31"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,2</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="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a>, <a class="txt" href="matrix_9.html#E3:31"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="matrix_9.html#E4:31"><i><font color="Green" title="E27">A4</font></i></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="E28:31"/><i><font color="Green" title="E45">A22</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> 3,2 <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> 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="E29:31"/><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c15">z</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="c9">h</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E25:31"><i><font color="Green" title="E44">A21</font></i></a>, <a class="txt" href="matrix_9.html#E28:31"><i><font color="Green" title="E45">A22</font></i></a></i>;<br/>
<a NAME="E30:31"/>
<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="c9">h</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E11:31"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="matrix_9.html#E12:31"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="matrix_3.html#D7" title="MATRIX_3:def.7">MATRIX_3:def 7</a>, <a class="txt" href="matrix_9.html#E29:31"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="matrix_9.html#E28:31"><i><font color="Green" title="E45">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E31:31"/>
 <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="c2">a</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">h</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#E18:31"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="matrix_9.html#E24:31"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a>, <a class="txt" href="matrix_9.html#E5:31"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>


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