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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:3_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E2:3_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E3:3_1"/><i><font color="Green" title="E6">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">B</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">B</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_1_1"/><i><font color="Green" title="E7">A6</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font>] )
 ;<br/>

<a NAME="E2:3_1_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c1">x</font> as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>
<a NAME="E4:3_1_1"/><i><font color="Green" title="E9">A8</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="matrix_9.html#E2:3_1_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:3_1_1"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c2">B</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> or ( not <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> ) or <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1_1"/><i><font color="Green" title="E11">A10</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="matrix_9.html#E4:3_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="matrix_9.html#E5:3_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E3:3_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:3_1_1_1_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_1_2"/><i><font color="Green" title="E11">A11</font></i>: 
( not <font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <b>by </b><i><a class="txt" href="matrix_9.html#E4:3_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="txt" href="matrix_9.html#E5:3_1_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E3:3_1_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="matrix_9.html#E2:3"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="matrix_9.html#E1:3_1_1_1_2"><i><font color="Green" title="E11">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1_3"><b>suppose </b></a><a NAME="E1:3_1_1_1_3"/>
<font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:3_1_1_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:3"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="matrix_9.html#E1:3_1_1"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">B</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 
<b>from </b><i><a class="ref" href="finset_1.html#S2" title="FINSET_1:sch.2">FINSET_1:sch 2</a>(<a class="txt" href="matrix_9.html#E1:3_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="txt" href="matrix_9.html#E2:3_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="txt" href="matrix_9.html#E3:3_1"><i><font color="Green" title="E6">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 ;<br/>


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