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<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> holds   <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default">(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="b4">d</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<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="b3">c</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">d</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="c3">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">c</font></span>)</span></span><br/>









<span class="kw">reconsider </span><font color="Maroon" title="c6">rid2</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> 2</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> 2 <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">set </span><font color="Maroon" title="c7">M</font> = (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">Id2</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> 2 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> 2 <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="c9">id2</font> = <font color="Maroon" title="c8">Id2</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> 2</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">f</font> =  <a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>;<br/>
<a NAME="E4:22"/><span class="lab"><font color="Green" title="E14">A0</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> 2
 
;<br/>
<a NAME="E5:22"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> 2 <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> 2</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="E6:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A1</font></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> 2</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> 2</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> 2 &amp; <font color="Maroon" title="c9">id2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">rid2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E14">A0</font></span></a>, <a class="ref" href="matrix_9.html#T2" target="_self" onmouseover="rs('matrix_9/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a>, <a class="ref" href="subset_1.html#D8" onmouseover="rs('subset_1/D8')" onmouseout="rh()">SUBSET_1:def 8</a></span>;<br/>
<span class="lab"><font color="Green" title="E16">A2</font></span>: <span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">rid2</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c6">rid2</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T10" target="_self" onmouseover="rs('matrix_9/T10')" onmouseout="rh()">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T11" target="_self" onmouseover="rs('matrix_9/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">id2</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">$$</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c8">Id2</font>,<span class="p2">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#D1" target="_self" onmouseover="rs('matrix_9/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T9" target="_self" onmouseover="rs('matrix_9/T9')" onmouseout="rh()">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T11" target="_self" onmouseover="rs('matrix_9/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:22"/>
 <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="matrix_0.html#K15" title="MATRIX_0:func.15">][</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">d</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="c3">b</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c4">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E6:22"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="matrix_9.html#T6" target="_self" onmouseover="rs('matrix_9/T6')" onmouseout="rh()">Th6</a>, <a class="ref" href="setwop_2.html#T1" onmouseover="rs('setwop_2/T1')" onmouseout="rh()">SETWOP_2:1</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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