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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c2">P</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">N</font> = <font color="Maroon" title="c1">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>;<br/>
<b>set </b><font color="Maroon" title="c5">PER</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span> ;<br/>
<a NAME="E2:7"/><i><font color="Green" title="E7">A1</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span>  <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" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
 ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> )
 
;<br/>
<b>hence </b><a NAME="E3:7_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</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><b>end;</b></div>
<a NAME="E3:7"/>
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:7_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">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/>
<b>hence </b><a NAME="E3:7_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7"/><b>then </b>
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">F</font> where <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font>, <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> : <font color="Olive" title="b1">F</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="laplace.html#E2:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b> <a href="card_1.html#K1" title="CARD_1:func.1">Card</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> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2" title="SETWOP_2:func.2">Seg</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="newton.html#K12" title="NEWTON:func.12">!</a> 
<b>by </b><i><a class="ref" href="card_fin.html#T8" title="CARD_FIN:th.8">CARD_FIN:8</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="newton.html#K6" title="NEWTON:func.6">!</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a></i>
;<br/><br/>


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