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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> implies <font color="Maroon" title="c2">H</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:99"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:99"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">H</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font><b> such that </b><br/><a NAME="E4:99"/><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_6.html#E1:99"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="group_6.html#D15" target="_self" title="GROUP_6:def.15">Def15</a></i>;<br/>
<a NAME="E5:99"/>
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="group_6.html#E4:99"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="group_6.html#D14" target="_self" title="GROUP_6:def.14">Def14</a></i>;<br/>
<a NAME="E6:99"/><b>then </b><i><font color="Green" title="E67">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_6.html#D13" target="_self" title="GROUP_6:def.13">Def13</a></i>;<br/>
<a NAME="E7:99"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="group_6.html#E2:99"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>;<br/>
<a NAME="E8:99"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:99"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="group_6.html#E6:99"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<b>hence </b><a NAME="E10:99"/>
<font color="Maroon" title="c2">H</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D11" title="STRUCT_0:def.11">STRUCT_0:def 11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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