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<div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>thus </b><a NAME="E1:73"/>
( <font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a> implies  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<a NAME="E2:73_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E3:73_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b>
(  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R2">are_equipotent</a>  &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:73_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_1.html#T81">CARD_1:81</a></i>;<br/>
<b>hence </b><a NAME="E6:73_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:73"/><i><font color="Green">E32</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">p</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>, <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green">E33</font></i>: 
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E6:73"/>
(  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:73"/>
<font color="Maroon">p</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="finseq_4.html#T7" target="_self">Th7</a></i>;<br/>


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