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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/> <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Olive" title="b10">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b5">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b9">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b3">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b5">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b7">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b6">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b8">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b6">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b7">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b9">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b8">h</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c2">rid3</font> =  <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> as    <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 <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">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font>, <font color="Maroon" title="c7">e</font>, <font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font>, <font color="Maroon" title="c11">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="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">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="c6">d</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c8">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="c9">g</font>,<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c11">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/> <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c12">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="c12">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="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">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="c6">d</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c8">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="c9">g</font>,<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c11">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  <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E2:61"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c12">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="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">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="c6">d</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c8">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="c9">g</font>,<font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c11">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">  <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">a3</font> = <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>, <font color="Maroon" title="c14">a4</font> = <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>, <font color="Maroon" title="c15">a5</font> = <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>, <font color="Maroon" title="c16">a6</font> = <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> as    <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 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T27" target="_self" onmouseover="rs('matrix_9/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">id3</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 3 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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">Id3</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 3 as    <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 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">B1</font> = <span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c18">Id3</font>,<font color="Maroon" title="c2">rid3</font>,<font color="Maroon" title="c13">a3</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span>, <font color="Maroon" title="c20">B2</font> = <span class="p1"><a href="setwiseo.html#K4" title="SETWISEO:func.4">{.</a><span class="default"><font color="Maroon" title="c14">a4</font>,<font color="Maroon" title="c15">a5</font>,<font color="Maroon" title="c16">a6</font></span><a href="setwiseo.html#K4" title="SETWISEO:func.4">.}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">r</font> =  <a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font>;<br/>
<span class="lab"><font color="Green" title="E56">A2</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">id3</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c18">Id3</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</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">a</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c11">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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_9.html#T20" target="_self" onmouseover="rs('matrix_9/T20')" onmouseout="rh()">Th20</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E57">A3</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a6</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c16">a6</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</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">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c10">h</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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_9.html#T25" target="_self" onmouseover="rs('matrix_9/T25')" onmouseout="rh()">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E58">A4</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a5</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c15">a5</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c11">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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_9.html#T24" target="_self" onmouseover="rs('matrix_9/T24')" onmouseout="rh()">Th24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A5</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a4</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c14">a4</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_9.html#T23" target="_self" onmouseover="rs('matrix_9/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_5"><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> <font color="Maroon" title="c19">B1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">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="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font> implies <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:61_5"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</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="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font></span><br/><a NAME="E2:61_5"/><span class="kw">then </span>
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">Id3</font> or <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">rid3</font> or <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a3</font> )
 <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></span>;<br/><a NAME="E3:61_5"/><span class="kw">then </span>
 not <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="matrix_9.html#E34"><span class="lab"><font color="Green" title="E28">Lm5</font></span></a>, <a class="txt" href="matrix_9.html#E35"><span class="lab"><font color="Green" title="E29">Lm6</font></span></a>, <a class="txt" href="matrix_9.html#E36"><span class="lab"><font color="Green" title="E30">Lm7</font></span></a>, <a class="txt" href="matrix_9.html#E37"><span class="lab"><font color="Green" title="E31">Lm8</font></span></a>, <a class="txt" href="matrix_9.html#E38"><span class="lab"><font color="Green" title="E32">Lm9</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="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_5"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_5"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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="E12:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c19">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font>
 
<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/>
<a NAME="E13:61"/>
 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> <font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E14:61"/><span class="kw">then </span>
<font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">B1</font>
 
<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/>
<a NAME="E15:61"/><span class="kw">then </span>
<font color="Maroon" title="c19">B1</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c20">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61"><span class="lab"><font color="Green" title="E60">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>;<br/>
<a NAME="E16:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<font color="Maroon" title="c19">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c20">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T83" onmouseover="rs('xboole_1/T83')" onmouseout="rh()">XBOOLE_1:83</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">F</font> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font>;<br/>
<a NAME="E17:61"/>
<font color="Maroon" title="c17">id3</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="ref" href="matrix_1.html#D12" onmouseover="rs('matrix_1/D12')" onmouseout="rh()">MATRIX_1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">r1</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">id3</font>, <font color="Maroon" title="c24">r2</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">rid3</font>, <font color="Maroon" title="c25">r3</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a3</font>, <font color="Maroon" title="c26">r4</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">a4</font>, <font color="Maroon" title="c27">r5</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">a5</font>, <font color="Maroon" title="c28">r6</font> = <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">a6</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/>
<a NAME="E19:61"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" onmouseover="rs('finsub_1/D5')" onmouseout="rh()">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E20:61"/><span class="kw">then </span>
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="subset_1.html#D8" onmouseover="rs('subset_1/D8')" onmouseout="rh()">SUBSET_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">X</font> = <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><font color="Maroon" title="c18">Id3</font>,<font color="Maroon" title="c2">rid3</font>,<font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c14">a4</font>,<font color="Maroon" title="c15">a5</font>,<font color="Maroon" title="c16">a6</font></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T15" target="_self" onmouseover="rs('matrix_9/T15')" onmouseout="rh()">Th15</a>, <a class="ref" href="matrix_9.html#T19" target="_self" onmouseover="rs('matrix_9/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/>
<a NAME="E22:61"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
(  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c19">B1</font>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c24">r2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c25">r3</font> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c20">B2</font>,<span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r4</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c27">r5</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c28">r6</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrix_9.html#E32"><span class="lab"><font color="Green" title="E26">Lm3</font></span></a>, <a class="txt" href="matrix_9.html#E33"><span class="lab"><font color="Green" title="E27">Lm4</font></span></a>, <a class="txt" href="matrix_9.html#E39"><span class="lab"><font color="Green" title="E33">Lm10</font></span></a>, <a class="txt" href="matrix_9.html#E40"><span class="lab"><font color="Green" title="E34">Lm11</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="setwop_2.html#T3" onmouseover="rs('setwop_2/T3')" onmouseout="rh()">SETWOP_2:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E63">A10</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">rid3</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c2">rid3</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</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">c</font>,<font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c9">g</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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</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="matrix_9.html#T21" target="_self" onmouseover="rs('matrix_9/T21')" onmouseout="rh()">Th21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<span class="lab"><font color="Green" title="E64">A11</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">a3</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c13">a3</font>,<font color="Maroon" title="c12">M</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</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">a</font>,<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c10">h</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="#E2:61"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="matrix_9.html#T22" target="_self" onmouseover="rs('matrix_9/T22')" onmouseout="rh()">Th22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T26" target="_self" onmouseover="rs('matrix_9/T26')" onmouseout="rh()">Th26</a></span>
;<br/>
<a NAME="E25:61"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 3</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">X</font> &amp; <font color="Maroon" title="c29">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c18">Id3</font>,<font color="Maroon" title="c2">rid3</font>,<font color="Maroon" title="c13">a3</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c14">a4</font>,<font color="Maroon" title="c15">a5</font>,<font color="Maroon" title="c16">a6</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T15" target="_self" onmouseover="rs('matrix_9/T15')" onmouseout="rh()">Th15</a>, <a class="ref" href="matrix_9.html#T19" target="_self" onmouseover="rs('matrix_9/T19')" onmouseout="rh()">Th19</a>, <a class="ref" href="enumset1.html#T13" onmouseover="rs('enumset1/T13')" onmouseout="rh()">ENUMSET1:13</a>, <a class="ref" href="finseq_2.html#T53" onmouseover="rs('finseq_2/T53')" onmouseout="rh()">FINSEQ_2:53</a></span>;<br/>
<span class="kw">then </span> <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Maroon" title="c12">M</font> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c19">B1</font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c20">B2</font>,<span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <font color="Maroon" title="c12">M</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="setwop_2.html#T4" onmouseover="rs('setwop_2/T4')" onmouseout="rh()">SETWOP_2:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c24">r2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c25">r3</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">r4</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">r5</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c28">r6</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">r1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c24">r2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c25">r3</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c26">r4</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">r5</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c28">r6</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E23:61"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E24:61"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E10:61"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E9:61"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E8:61"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E27:61"/>
 <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <font color="Maroon" title="c12">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">g</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c11">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">d</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">h</font></span>)</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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