<?xml version="1.0"?>
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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>, <font color="Olive" title="b5">e</font>, <font color="Olive" title="b6">f</font>, <font color="Olive" title="b7">g</font>, <font color="Olive" title="b8">h</font>, <font color="Olive" title="b9">i</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="b10">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b10">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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="Olive" title="b4">d</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b6">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="Olive" title="b7">g</font>,<font color="Olive" title="b8">h</font>,<font color="Olive" title="b9">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> holds <br/> for <font color="Olive" title="b11">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> 3  st <font color="Olive" title="b11">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b11">p</font>,<font color="Olive" title="b10">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="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b9">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&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>, <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>; <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">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">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> holds <br/> for <font color="Olive" title="b2">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> 3  st <font color="Olive" title="b2">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b2">p</font>,<font color="Olive" title="b1">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">M</font> be   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<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"> ( <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> implies  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> 3  st <font color="Olive" title="b1">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b1">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )</span><br/>





<a NAME="E1:32"/>
<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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T31" onmouseover="rs('matrix_0/T31')" onmouseout="rh()">MATRIX_0:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E3:32"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c12">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
 <span class="kw">and </span><br/><a NAME="E4:32"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D5" onmouseover="rs('matrix_0/D5')" onmouseout="rh()">MATRIX_0:def 5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:32"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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>
 ; <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> 3  st <font color="Olive" title="b1">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> holds <br/> <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Olive" title="b1">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/>

<a NAME="E6:32"/><span class="kw">then </span>
<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>
 
;<br/>
<a NAME="E7:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E23">A1</font></span></a></span>;<br/>
<a NAME="E8:32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,3</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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T31" onmouseover="rs('matrix_0/T31')" onmouseout="rh()">MATRIX_0:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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><span class="kw"> such that </span><br/><a NAME="E10:32"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c13">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
 <span class="kw">and </span><br/><a NAME="E11:32"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (3,3) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D5" onmouseover="rs('matrix_0/D5')" onmouseout="rh()">MATRIX_0:def 5</a></span>;<br/>
<a NAME="E12:32"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E13:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c13">z</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="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<a NAME="E14:32"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</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_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c11">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T31" onmouseover="rs('matrix_0/T31')" onmouseout="rh()">MATRIX_0:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E16:32"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<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
 <span class="kw">and </span><br/><a NAME="E17:32"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c11">M</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (2,1) <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> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D5" onmouseover="rs('matrix_0/D5')" onmouseout="rh()">MATRIX_0:def 5</a></span>;<br/>
<a NAME="E18:32"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E19:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c14">y</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="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">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> 3; <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="c15">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> implies  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c15">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E20:32"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c15">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">2,1,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</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">  <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c15">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/>

<a NAME="E21:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A12</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
;<br/>
<a NAME="E22:32"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>;<br/>
<a NAME="E23:32"/><span class="lab"><font color="Green" title="E36">A14</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="#E20:32"><span class="lab"><font color="Green" title="E33">A11</font></span></a></span>;<br/>
<a NAME="E24:32"/><span class="lab"><font color="Green" title="E37">A15</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="c15">p</font>,<font color="Maroon" title="c11">M</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E25:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A16</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="c15">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
 
<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="E26:32"/><span class="kw">then </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="c15">p</font>,<font color="Maroon" title="c11">M</font>)</span>)</span>
 
;<br/>
<a NAME="E27:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A17</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="c15">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="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E17:32"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E19:32"><span class="lab"><font color="Green" title="E32">A10</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/>
<a NAME="E28:32"/>
3 <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="c15">p</font>,<font color="Maroon" title="c11">M</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/>
<a NAME="E29:32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A18</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="c15">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="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:32"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E11:32"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E13:32"><span class="lab"><font color="Green" title="E29">A7</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/>
<a NAME="E30:32"/>
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="c15">p</font>,<font color="Maroon" title="c11">M</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:32"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/>
<a NAME="E31:32"/><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="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E7:32"><span class="lab"><font color="Green" title="E26">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/>
<span class="kw">hence </span><a NAME="E32:32"/>
 <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c15">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="c5">d</font>,<font color="Maroon" title="c10">i</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:32"><span class="lab"><font color="Green" title="E37">A15</font></span></a>, <a class="txt" href="#E27:32"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E29:32"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</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/>


</div>
