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

<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">n</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">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Olive" title="b1">n</font>,<font color="Olive" title="b1">n</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="c1">K</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <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> ; <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="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</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="c1">K</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">B</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="c2">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="c2">n</font></span>)</span></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  <a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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/>
<span class="kw">set </span><font color="Maroon" title="c6">Y</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">X</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">G0</font> = <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="c2">n</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c2">n</font></span>)</span></span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> ;<br/>
<a NAME="E2:14"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>  st <font color="Olive" title="b1">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> holds <br/><font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> ( <font color="Maroon" title="c9">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> implies <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font> )</span><br/>

<a NAME="E1:14_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fvsum_1.html#T6" onmouseover="rs('fvsum_1/T6')" onmouseout="rh()">FVSUM_1:6</a></span>;<br/>
<a NAME="E2:14_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</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="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c9">e</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:14_1"/>
<font color="Maroon" title="c9">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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"> <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font></span><br/>

<a NAME="E4:14_1"/><span class="kw">then </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="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>,<font color="Maroon" title="c9">e</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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:14_1"/>
<font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="finsub_1.html#T7" onmouseover="rs('finsub_1/T7')" onmouseout="rh()">FINSUB_1:7</a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:14"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</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="c1">K</font></span><br/>

<a NAME="E4:14"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 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>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> holds <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> implies <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></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"> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E2:14_2"/>
 for <font color="Olive" title="b1">p</font> being    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> 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">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">p</font> be    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Element</a> of  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">$$</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="c10">p</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><br/>

<a NAME="E1:14_2_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<a NAME="E1:14_2_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T60" onmouseover="rs('finseq_2/T60')" onmouseout="rh()">FINSEQ_2:60</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:14_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">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">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span> 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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_2_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="finsop_1.html#T4" onmouseover="rs('finsop_1/T4')" onmouseout="rh()">FINSOP_1:4</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>

;<br/>
<span class="kw">hence </span><a NAME="E4:14_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:14_2_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font>, <font color="Maroon" title="c12">j</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"> ( <font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_2_1_2"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:14_2_1_2"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)</span><br/>

<a NAME="E3:14_2_1_2"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c11">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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_2_1_2"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E4:14_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">j</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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_2_1_2"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="ref" href="matrix_7.html#T14" onmouseover="rs('matrix_7/T14')" onmouseout="rh()">MATRIX_7:14</a></span>;<br/>
<a NAME="E5:14_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c12">j</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> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/>
<a NAME="E6:14_2_1_2"/>
<font color="Maroon" title="c11">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_2_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="matrix_7.html#T13" onmouseover="rs('matrix_7/T13')" onmouseout="rh()">MATRIX_7:13</a></span>;<br/>
<a NAME="E7:14_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p4">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_2_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<a NAME="E8:14_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D4" onmouseover="rs('matrix_0/D4')" onmouseout="rh()">MATRIX_0:def 4</a></span>;<br/>
<a NAME="E9:14_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14_2_1_2"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:14_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c11">i</font>,<font color="Maroon" title="c12">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_2_1_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="matrix_3.html#T1" onmouseover="rs('matrix_3/T1')" onmouseout="rh()">MATRIX_3:1</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/>


</div><span class="kw">end;</span></div>
<a NAME="E3:14_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" onmouseover="rs('card_1/D7')" onmouseout="rh()">CARD_1:def 7</a></span>;<br/>
<a NAME="E4:14_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 <a href="matrix_3.html#K10" title="MATRIX_3:func.10">Path_matrix</a> (<font color="Maroon" title="c10">p</font>,<span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_2_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="ref" href="matrix_3.html#D7" onmouseover="rs('matrix_3/D7')" onmouseout="rh()">MATRIX_3:def 7</a></span>;<br/>
<a NAME="E5:14_2_1"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:14"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="xreal_1.html#T235" onmouseover="rs('xreal_1/T235')" onmouseout="rh()">XREAL_1:235</a></span>;<br/>
<a NAME="E6:14_2_1"/>
1 <a href="finseq_2.html#K5" title="FINSEQ_2:func.5">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T59" onmouseover="rs('finseq_2/T59')" onmouseout="rh()">FINSEQ_2:59</a></span>;<br/>
<a NAME="E7:14_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsop_1.html#T11" onmouseover="rs('finsop_1/T11')" onmouseout="rh()">FINSOP_1:11</a></span>;<br/>
<a NAME="E8:14_2_1"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E7:14_2_1"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E1:14_2_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>);<br/></span>
<a NAME="E9:14_2_1"/><span class="kw">then </span>
 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="c10">p</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_2_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E5:14_2_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:14_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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="c10">p</font>,<span class="p2">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:14_2"/>
<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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:14"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<a NAME="E6:14"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A16</font></span>: 
 <a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/>
<a NAME="E8:14"/><span class="lab"><font color="Green" title="E14">A17</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="c2">n</font> holds  <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c2">n</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="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font></span><br/>

<a NAME="E1:14_3"/>
<font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></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="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:14_3"/>
<font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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="E9:14"/><span class="lab"><font color="Green" title="E15">A18</font></span>: 
 for <font color="Olive" title="b1">B9</font> being    <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="c2">n</font></span>)</span>  st <font color="Olive" title="b1">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c2">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="c2">n</font></span>)</span></span>)</span>) &amp; <font color="Olive" title="b1">B9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">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="c2">n</font>  st <font color="Olive" title="b2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Olive" title="b1">B9</font> holds <br/><font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">B9</font> be    <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="c2">n</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="c9">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c2">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="c2">n</font></span>)</span></span>)</span>) &amp; <font color="Maroon" title="c9">B9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  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="c2">n</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c9">B9</font> holds <br/><font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_4"/>
<font color="Maroon" title="c9">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c2">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="c2">n</font></span>)</span></span>)</span>)
 <span class="kw">and </span><br/><a NAME="E2:14_4"/>
<font color="Maroon" title="c9">B9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">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="c2">n</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c9">B9</font> holds <br/><font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>)</span><br/>

<span class="kw">thus </span><a NAME="E3:14_4"/>
 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="c2">n</font>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c9">B9</font> holds <br/><font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c2">n</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="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c9">B9</font> implies <font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_4_1"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c2">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="c2">n</font></span>)</span></span>)</span>)</span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c9">B9</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="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>)</span><br/>

<a NAME="E2:14_4_1"/><span class="lab"><font color="Green" title="E16">A19</font></span>: 
( <font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</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="c10">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></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="c1">K</font> &amp; <font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</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="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<a NAME="E3:14_4_1"/>
( <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"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</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="c1">K</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14"><span class="lab"><font color="Green" title="E13">A16</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a>, <a class="ref" href="fvsum_1.html#T6" onmouseover="rs('fvsum_1/T6')" onmouseout="rh()">FVSUM_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_4_1"/>
<font color="Maroon" title="c8">G0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B9</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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"><font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">B9</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_9.html#K1" title="MATRIX_9:func.1">PPath_product</a> <span class="p3">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_4_1"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="ref" href="binop_1.html#T3" onmouseover="rs('binop_1/T3')" onmouseout="rh()">BINOP_1:3</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>


</div><span class="kw">end;</span></div>
<a NAME="E10:14"/>
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Permutations</a> <font color="Maroon" title="c2">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="c2">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="E11:14"/><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="c2">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="c2">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="c2">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/>
<a NAME="E12:14"/><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="c2">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="c2">n</font></span>)</span></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">G0</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="c2">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="c2">n</font></span>)</span></span>)</span>)</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="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:14"/>
 <a href="matrix_9.html#K2" title="MATRIX_9:func.2">Per</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K1" title="MATRIX_3:func.1">0.</a> (<font color="Maroon" title="c1">K</font>,<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c2">n</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="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:14"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="txt" href="#E9:14"><span class="lab"><font color="Green" title="E15">A18</font></span></a>, <a class="ref" href="setwiseo.html#D3" onmouseover="rs('setwiseo/D3')" onmouseout="rh()">SETWISEO:def 3</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>
