<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:63"/>
(  <a href="group_1.html#K8">Ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="group_1.html#K8">Ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K8">Ord</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="group_1.html#K8">Ord</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="group_3.html#T77" target="_self">Th77</a></i>;<br/>
<a NAME="E2:63"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E3:63"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> iff the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>hence </b><a NAME="E4:63"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="group_1.html#V6">finite</a> iff <font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a> )
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>


</div>
