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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">K</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b2">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st  ex <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b4">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> ) ) holds <br/> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b2">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st  ex <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) ) holds <br/> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Olive" title="b1">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">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"> (  ex <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> ) ) implies  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c3">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">n1</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">M1</font> = <font color="Maroon" title="c3">M</font> as   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c4">n1</font>,<font color="Maroon" title="c2">K</font> ;<br/>
<span class="kw">given </span><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E4:65"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
( <font color="Maroon" title="c6">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">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">  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c3">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">P1</font> =  <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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f</font> =  <a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">F</font> =  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font>;<br/>
<a NAME="E5:65"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font> <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> <font color="Maroon" title="c1">n</font></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="E6:65"/><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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c1">n</font>
 
<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="c10">P1</font> =  <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> <font color="Maroon" title="c1">n</font></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> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> <font color="Maroon" title="c4">n1</font></span>)</span> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> <font color="Maroon" title="c4">n1</font></span>)</span>] <span class="kw">means </span> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> ($1,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>;<br/>
<a NAME="E8:65"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c4">n1</font><br/>  for <font color="Olive" title="b2">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> <font color="Maroon" title="c4">n1</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font> &amp;  not <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c4">n1</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">B</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> <font color="Maroon" title="c4">n1</font></span>)</span>  st <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font> &amp;  not <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">B</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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> <font color="Maroon" title="c4">n1</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"> ( <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font> &amp; <font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font> &amp;  not <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">B</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:65_1"/>
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">and </span><br/><a NAME="E2:65_1"/>
<font color="Maroon" title="c12">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">and </span><br/><a NAME="E3:65_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 not <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">B</font>
 <span class="kw">and </span><br/><a NAME="E4:65_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">B</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">S<sub>1</sub></font>[<font color="Maroon" title="c12">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c12">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span>) = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c12">B</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="setwop_2.html#T2" onmouseover="rs('setwop_2/T2')" onmouseout="rh()">SETWOP_2:2</a></span>
<br/>.= 
<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="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<font color="Maroon" title="c12">B</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span>)</span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="matrix_9.html#T50" target="_self" onmouseover="rs('matrix_9/T50')" onmouseout="rh()">Th50</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="rlvect_1.html#T4" onmouseover="rs('rlvect_1/T4')" onmouseout="rh()">RLVECT_1:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:65_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">B</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.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="E9:65"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c4">n1</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c4">n1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</font> implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_2"/>
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">P1</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">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]</span><br/>

 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c5">M1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#T17" onmouseover="rs('setwiseo/T17')" onmouseout="rh()">SETWISEO:17</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="matrix_9.html#T50" target="_self" onmouseover="rs('matrix_9/T50')" onmouseout="rh()">Th50</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:65_2"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c11">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.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="E10:65"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">P1</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="matrix_9.html#S1" target="_self" onmouseover="rs('matrix_9/S1')" onmouseout="rh()">MATRIX_9:sch 1</a>(<a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E60">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E11:65"/>
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">K</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> (<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_3.html#K11" title="MATRIX_3:func.11">Path_product</a> <font color="Maroon" title="c3">M</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">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"> verum</span><br/>


</div>
