<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="matrix_1.html#NM5">Matrix</a> of 2,<font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:12"/><i><font color="Green">E15</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="finseq_1.html#K2">Seg</a> 2 as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="matrix_7.html#E1:12"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">1,2</span><a href="finseq_1.html#K10">*&gt;</a></span>, <font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default">2,1</span><a href="finseq_1.html#K10">*&gt;</a></span> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> 2 <b>by </b><i><a class="ref" href="matrix_7.html#T2" target="_self">Th2</a></i>;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:12_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:12_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green">E17</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E8:12"/><i><font color="Green">E19</font></i>: 
 <a href="matrix_2.html#K11">Permutations</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:12"/><i><font color="Green">E20</font></i>: 
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></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#K11">Permutations</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D9">MATRIX_3:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="matrix_2.html#K11">Permutations</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E10:12"/><i><font color="Green">E21</font></i>: 
 <a href="matrix_2.html#K11">Permutations</a> 2 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="E11:12"/><i><font color="Green">E22</font></i>: 
 <a href="matrix_2.html#K11">Permutations</a> 2 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E8:12"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E12:12"/><i><font color="Green">E23</font></i>: 
 <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K11">Permutations</a> 2
 
<b>by </b><i><a class="txt" href="matrix_7.html#E10:12"><i><font color="Green">E21</font></i></a>, <a class="ref" href="matrix_2.html#D17">MATRIX_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:12"/><i><font color="Green">E24</font></i>: 
(  <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</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#K11">Permutations</a> 2</span>)</span></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 holds <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <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#K11">Permutations</a> 2</span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</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#K11">Permutations</a> 2</span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E9:12"><i><font color="Green">E20</font></i></a>, <a class="ref" href="setwiseo.html#D3">SETWISEO:def 3</a></i>;<br/>
<i><font color="Green">E25</font></i>:  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span> = 
 <a href="matrix_2.html#K11">Permutations</a> 2
<b>by </b><i><a class="txt" href="matrix_7.html#E12:12"><i><font color="Green">E23</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="txt" href="matrix_7.html#E11:12"><i><font color="Green">E22</font></i></a></i>
;<br/>
<a NAME="E16:12"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> holds <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E11:12"><i><font color="Green">E22</font></i></a>, <a class="txt" href="matrix_7.html#E14:12"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E17:12"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E18:12"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E16:12"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E19:12"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</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#K11">Permutations</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="setwiseo.html#K2">.}</a></span> 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#K11">Permutations</a> 2</span>)</span> ;<br/>
<a NAME="E2:12_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E2:12_3_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K15">FinOmega</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E12:12"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<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#K11">Permutations</a> 2</span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="matrix_7.html#E15:12"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="txt" href="matrix_7.html#E4:12"><i><font color="Green">E16</font></i></a></i>
;<br/>
<a NAME="E4:12_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</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#K11">Permutations</a> 2</span>)</span></span>)</span> <a href="finsub_1.html#K2">\</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:12_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E14:12"><i><font color="Green">E24</font></i></a>, <a class="txt" href="matrix_7.html#E2:12_3"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:12_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</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#K11">Permutations</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E18:12"><i><font color="Green">E27</font></i></a>, <a class="txt" href="matrix_7.html#E15:12"><i><font color="Green">E25</font></i></a>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:12"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>;<br/>
<a NAME="E21:12"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">"**"</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D8">MATRIX_3:def 8</a></i>;<br/>
<a NAME="E22:12"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E23:12"/><i><font color="Green">E32</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
;<br/>
<a NAME="E24:12"/><b>then </b><i><font color="Green">E33</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E6:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:12"/><i><font color="Green">E34</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
;<br/>
<a NAME="E26:12"/><b>then </b><i><font color="Green">E35</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E6:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E27:12"/><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E23:12"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_7.html#E7:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:12"/><i><font color="Green">E37</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E25:12"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrix_7.html#E7:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E29:12"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E30:12"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E31:12"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,1
 
<b>by </b><i><a class="txt" href="matrix_7.html#E24:12"><i><font color="Green">E33</font></i></a>, <a class="txt" href="matrix_7.html#E22:12"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E32:12"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E33:12"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,2
 
<b>by </b><i><a class="txt" href="matrix_7.html#E26:12"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrix_7.html#E22:12"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E34:12"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E35:12"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,2
 
<b>by </b><i><a class="txt" href="matrix_7.html#E27:12"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrix_7.html#E29:12"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E36:12"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E37:12"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,1
 
<b>by </b><i><a class="txt" href="matrix_7.html#E28:12"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrix_7.html#E29:12"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E39:12"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 holds <br/><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> 2 )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="txt" href="matrix_7.html#E6:12"><i><font color="Green">E17</font></i></a></i>;<br/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E39:12"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,2</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a>, <a class="txt" href="matrix_7.html#E31:12"><i><font color="Green">E39</font></i></a>, <a class="txt" href="matrix_7.html#E39:12"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_7.html#E33:12"><i><font color="Green">E40</font></i></a></i>
;<br/>
<a NAME="E41:12"/><b>then </b><i><font color="Green">E44</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,2</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E39:12"><i><font color="Green">E43</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E43:12"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 holds <br/><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> 2 )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="txt" href="matrix_7.html#E7:12"><i><font color="Green">E18</font></i></a></i>;<br/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E43:12"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a>, <a class="txt" href="matrix_7.html#E35:12"><i><font color="Green">E41</font></i></a>, <a class="txt" href="matrix_7.html#E43:12"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrix_7.html#E37:12"><i><font color="Green">E42</font></i></a></i>
;<br/>
<a NAME="E45:12"/><b>then </b><i><font color="Green">E46</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E43:12"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E46:12"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E20:12"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E47:12"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E21:12"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E48:12"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E8:12"><i><font color="Green">E19</font></i></a>, <a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K2">}</a></span> as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span> <b>by </b><i><a class="txt" href="matrix_7.html#E48:12"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E50:12"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E51:12"/><i><font color="Green">E51</font></i>: 
0 <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>;<br/>
<i><font color="Green">E52</font></i>:  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>19</sub></font> = 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span></span>)</span>

<br/>.= 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="matrix_7.html#E4"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="matrix_2.html#T28">MATRIX_2:28</a></i>
;<br/>
<a NAME="E53:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 st <br/>( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E50:12"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E54:12"/><b>then </b><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span> st <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  ) ) )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E50:12"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrix_7.html#E51:12"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_7.html#E52:12"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E55:12"/><i><font color="Green">E54</font></i>: 
 <a href="matrix_2.html#K10">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/>
<a NAME="E56:12"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D15">MATRIX_2:def 15</a>, <a class="txt" href="matrix_7.html#E54:12"><i><font color="Green">E53</font></i></a></i>;<br/>
<i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="matrix_7.html#E46:12"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E56:12"><i><font color="Green">E55</font></i></a>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
;<br/>
<a NAME="E58:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 2</span>)</span> holds <br/>(  not <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">b<sub>1</sub></font> or ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> 2 holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>3</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  ) ) ) )
 
<b>by </b><i><a class="ref" href="matrix_7.html#T11" target="_self">Th11</a>, <a class="txt" href="matrix_7.html#E10"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<a NAME="E59:12"/><b>then </b><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> is <a href="matrix_2.html#V3">even</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#D15">MATRIX_2:def 15</a>, <a class="txt" href="matrix_7.html#E55:12"><i><font color="Green">E54</font></i></a></i>;<br/>
<i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="matrix_2.html#K14">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="matrix_7.html#E47:12"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E59:12"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#D16">MATRIX_2:def 16</a></i>
;<br/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>2</sub></font> = 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11">Path_product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E14:12"><i><font color="Green">E24</font></i></a>, <a class="txt" href="matrix_7.html#E19:12"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default">the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finsop_1.html#K2">$$</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E57:12"><i><font color="Green">E56</font></i></a>, <a class="txt" href="matrix_7.html#E60:12"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,2</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E41:12"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E45:12"><i><font color="Green">E46</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E62:12"/>
 <a href="matrix_3.html#K12">Det</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,2</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 1,2</span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> 2,1</span>)</span></span>)</span>
 ;<br/>


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