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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">p</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b3">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/>

<a NAME="E2:34"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20" title="MATRIX_2:th.20">MATRIX_2:20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">pf</font> = <font color="Maroon" title="c3">p</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon" title="c3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> or  not <font color="Maroon" title="c3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:34_1_1"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>;<br/><a NAME="E3:34_1_1"/>
<font color="Maroon" title="c5">pf</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:34"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="matrix_7.html#E2:34"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="matrix_7.html#E1:34_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="matrix_7.html#T28" target="_self" title="MATRIX_7:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:34"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="matrix_7.html#E2:34_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green" title="E31">A5</font></i>: 
not <font color="Maroon" title="c3">p</font> is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:34_1_2"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>;<br/><a NAME="E3:34_1_2"/>
not <font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:34"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="matrix_7.html#E1:34_1_2"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="matrix_7.html#T28" target="_self" title="MATRIX_7:th.28">Th28</a></i>;<br/><b>hence </b><a NAME="E4:34_1_2"/>
 <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K14" title="MATRIX_2:func.14">-</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:34_1_2"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="matrix_2.html#D16" title="MATRIX_2:def.16">MATRIX_2:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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