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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font><br/>  for <font color="Olive" title="b5">p</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 2  st <font color="Olive" title="b5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b5">p</font>,<span class="p1">(<span class="default">(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 2  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 2; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 implies  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> )</span><br/>











<span class="kw">assume </span><a NAME="E1:15"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span></span><br/>

<a NAME="E2:15"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D7" onmouseover="rs('matrix_3/D7')" onmouseout="rh()">MATRIX_3:def 7</a></span>;<br/>
<a NAME="E3:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span>
 
;<br/>
<a NAME="E5:15"/><span class="kw">then </span>
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (2,2)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:15"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="matrix_3.html#D7" onmouseover="rs('matrix_3/D7')" onmouseout="rh()">MATRIX_3:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T50" onmouseover="rs('matrix_0/T50')" onmouseout="rh()">MATRIX_0:50</a></span>
;<br/>
<a NAME="E7:15"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<a NAME="E8:15"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="matrix_3.html#D7" onmouseover="rs('matrix_3/D7')" onmouseout="rh()">MATRIX_3:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T50" onmouseover="rs('matrix_0/T50')" onmouseout="rh()">MATRIX_0:50</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:15"/>
 <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">p</font>,<span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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