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<b>let </b><font color="Maroon">H</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>





<b>assume </b><a NAME="E1:69"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<a NAME="E2:69"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:69"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E4:69"><i><font color="Green">E51</font></i></a>, <a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">ci</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:69"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">ci</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">ci</font> )
 <b>by </b><i><a class="txt" href="group_6.html#E6:69"><i><font color="Green">E52</font></i></a>, <a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E7:69"/>
 <a href="group_1.html#K8">Ord</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="group_1.html#K8">Ord</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:69"/><b>then </b>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="group_1.html#K8">Ord</a> <font color="Maroon">G</font>
 
;<br/>
<a NAME="E9:69"/><b>then </b>
 <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
;<br/>
<b>hence </b><a NAME="E10:69"/>
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/>


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