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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <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#R2" title="GROUP_6:pred.2">are_isomorphic</a>  implies  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<b>assume </b><a NAME="E1:92"/><i><font color="Green" title="E59">A1</font></i>: 
<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R2" title="GROUP_6:pred.2">are_isomorphic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c2">H</font></span><br/>

<b>then 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="E3:92"/><i><font color="Green" title="E60">A2</font></i>: 
<font color="Maroon" title="c3">h</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="ref" href="group_6.html#D15" target="_self" title="GROUP_6:def.15">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">g1</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">H</font>,<font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:92"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c4">g1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="#E1:92"><i><font color="Green" title="E59">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="E6:92"/>
<font color="Maroon" title="c3">h</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="#E3:92"><i><font color="Green" title="E60">A2</font></i></a></i>;<br/>
<a NAME="E7:92"/><b>then </b>
 <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T67" target="_self" title="GROUP_6:th.67">Th67</a></i>;<br/>
<a NAME="E8:92"/><b>then </b><i><font color="Green" title="E62">A4</font></i>: 
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T61" target="_self" title="GROUP_6:th.61">Th61</a></i>;<br/>
<a NAME="E9:92"/>
<font color="Maroon" title="c4">g1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="txt" href="#E5:92"><i><font color="Green" title="E61">A3</font></i></a></i>;<br/>
<a NAME="E10:92"/><b>then </b>
 <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T67" target="_self" title="GROUP_6:th.67">Th67</a></i>;<br/>
<a NAME="E11:92"/><b>then </b>
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T61" target="_self" title="GROUP_6:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E12:92"/>
 <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="#E8:92"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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