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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">K</font> being   <a href="vectsp_1.html#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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>







<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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 <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D9" title="MATRIX_2:def.9">MATRIX_2:def 9</a></span>;<br/>
<a NAME="E3:34"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:34_1_1"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
(  <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> &amp; <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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix_7.html#T28" target="_self" title="MATRIX_7:th.28">Th28</a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_2"/>
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"/><span class="kw">then </span>
(  <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> &amp;  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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="matrix_7.html#T28" target="_self" title="MATRIX_7:th.28">Th28</a>, <a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></span>; <a class="txt" 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>

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