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<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:103_1"/>
(  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  )
 <b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma9</font></i></a></i>;<br/><a NAME="E3:103_1"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="card_1.html#T81">CARD_1:81</a>, <a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:103_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E2:103"/>
(  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="wellord2.html#R2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="finseq_1.html#E15"><i><font color="Green">Lemma9</font></i></a></i>;<br/>
<a NAME="E3:103"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" target="_self">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_1.html#T81">CARD_1:81</a>, <a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E4:103"/>
(  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


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