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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> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:80"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:80"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:80"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c3">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" target="_self" title="FINSEQ_4:th.73">Th73</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E1:80"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="finseq_4.html#E3:80"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="finseq_4.html#E4:80"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="finseq_4.html#T77" target="_self" title="FINSEQ_4:th.77">Th77</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">q</font> = <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <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> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#T19" title="FINSEQ_1:th.19">FINSEQ_1:19</a></i>;<br/>
<a NAME="E7:80"/><i><font color="Green" title="E48">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E5:80"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E8:80"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c4">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="finseq_4.html#E4:80"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">A</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="finseq_4.html#E3:80"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E10:80"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">A</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="finseq_4.html#E7:80"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="finseq_4.html#E8:80"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="finseq_4.html#T77" target="_self" title="FINSEQ_4:th.77">Th77</a></i>;<br/>
<b>hence </b><a NAME="E11:80"/>
 ex <font color="Olive" title="b1">A</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> st  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>


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