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

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 )</span><br/><a NAME="E1:27_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D12" onmouseover="rs('matrix_1/D12')" onmouseout="rh()">MATRIX_1:def 12</a></span>;<br/><a NAME="E2:27_1"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">h</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,2</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E2:27_1_1"/>
<span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">3</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">h</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">3,1,2</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="ref" href="finseq_1.html#T43" onmouseover="rs('finseq_1/T43')" onmouseout="rh()">FINSEQ_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27_1_1"/>
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="matrix_9.html#T18" target="_self" onmouseover="rs('matrix_9/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</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><span class="kw">end;</span></div><a NAME="E3:27_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">h</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">2,3</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#V1" title="FINSEQ_1:attr.1">FinSequence-like</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/>
<a NAME="E3:27_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">1</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" onmouseover="rs('finseq_1/T43')" onmouseout="rh()">FINSEQ_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:27_1_2"/>
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="matrix_9.html#T18" target="_self" onmouseover="rs('matrix_9/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</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><span class="kw">end;</span></div><a NAME="E4:27_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">h</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">2,3</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#V1" title="FINSEQ_1:attr.1">FinSequence-like</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/>
<a NAME="E3:27_1_3"/>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">h</font> <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">3,2</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="ref" href="finseq_5.html#T61" onmouseover="rs('finseq_5/T61')" onmouseout="rh()">FINSEQ_5:61</a></span>;<br/>
<a NAME="E4:27_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">1</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">h</font> &amp; <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">1</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T43" onmouseover="rs('finseq_1/T43')" onmouseout="rh()">FINSEQ_1:43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27_1_3"/>
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="matrix_9.html#T17" target="_self" onmouseover="rs('matrix_9/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</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><span class="kw">end;</span></div><a NAME="E5:27_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<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 href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">h</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,2</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<a NAME="E2:27_1_4"/>
<span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">3</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c2">h</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="matrix_9.html#T18" target="_self" onmouseover="rs('matrix_9/T18')" onmouseout="rh()">Th18</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/>
<a NAME="E3:27_1_4"/><span class="kw">then </span>
(  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">h</font> <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">2,1</span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default">3</span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T17" target="_self" onmouseover="rs('matrix_9/T17')" onmouseout="rh()">Th17</a>, <a class="ref" href="finseq_5.html#T61" onmouseover="rs('finseq_5/T61')" onmouseout="rh()">FINSEQ_5:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:27_1_4"/>
<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 href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T18" target="_self" onmouseover="rs('matrix_9/T18')" onmouseout="rh()">Th18</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><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E6:27_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</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"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span><br/><a NAME="E7:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T12" onmouseover="rs('enumset1/T12')" onmouseout="rh()">ENUMSET1:12</a></span>;<br/><a NAME="E8:27_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:27_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T5" onmouseover="rs('enumset1/T5')" onmouseout="rh()">ENUMSET1:5</a></span>;<br/><a NAME="E10:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A6</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E11:27_1"/>
 <a href="finseq_5.html#K3" title="FINSEQ_5:func.3">Rev</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 3</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T4" target="_self" onmouseover="rs('matrix_9/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E12:27_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E1:27_1"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E4:27_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E3:27_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:27_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="matrix_9.html#T15" target="_self" onmouseover="rs('matrix_9/T15')" onmouseout="rh()">Th15</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</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><span class="kw">end;</span></div>
<a NAME="E2:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">now </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="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 implies <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:27_2"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">q</font> = <font color="Maroon" title="c1">p</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D12" onmouseover="rs('matrix_1/D12')" onmouseout="rh()">MATRIX_1:def 12</a></span>;<br/><a NAME="E3:27_2"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c2">q</font> <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="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/><a NAME="E4:27_2"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 ;<br/><a NAME="E5:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E6:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A10</font></span>: 
( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" onmouseover="rs('finseq_3/T1')" onmouseout="rh()">FINSEQ_3:1</a></span>;<br/><a NAME="E7:27_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 ;<br/><a NAME="E8:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E9:27_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A12</font></span>: 
( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" onmouseover="rs('finseq_3/T1')" onmouseout="rh()">FINSEQ_3:1</a></span>;<br/><a NAME="E10:27_2"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c2">q</font> <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="funct_2.html#T52" onmouseover="rs('funct_2/T52')" onmouseout="rh()">FUNCT_2:52</a></span>;<br/><a NAME="E11:27_2"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
( (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) ) or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</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> or <font color="Maroon" title="c2">q</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c3">q</font> = <font color="Maroon" title="c2">q</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_2"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="finseq_1.html#D2" onmouseover="rs('finseq_1/D2')" onmouseout="rh()">FINSEQ_1:def 2</a></span>;<br/>
<a NAME="E2:27_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_2"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27_2_1"/>
( (  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) ) or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</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> or <font color="Maroon" title="c2">q</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">3,1,2</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="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><span class="kw">end;</span></div><a NAME="E12:27_2"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
 ;<br/><a NAME="E13:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="txt" href="#E3:27_2"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E14:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" onmouseover="rs('finseq_3/T1')" onmouseout="rh()">FINSEQ_3:1</a></span>;<br/><a NAME="E15:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c1">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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> or <font color="Maroon" title="c2">q</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> or <font color="Maroon" title="c2">q</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">3,1,2</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="#E10:27_2"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="txt" href="#E12:27_2"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="txt" href="#E7:27_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E4:27_2"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E9:27_2"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E6:27_2"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E11:27_2"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><a NAME="E16:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E17:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E18:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c2">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T5" onmouseover="rs('enumset1/T5')" onmouseout="rh()">ENUMSET1:5</a></span>;<br/><a NAME="E19:27_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,<span class="p2"><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>,<span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E20:27_2"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T12" onmouseover="rs('enumset1/T12')" onmouseout="rh()">ENUMSET1:12</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><span class="kw">end;</span></div>
<a NAME="E4:27"/><span class="kw">then </span>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:27"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default">1,2,3</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">3,2,1</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">1,3,2</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">2,3,1</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">2,1,3</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">3,1,2</span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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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