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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">H</font>,<font color="Maroon">I</font>;<br/>









<b>assume </b><a NAME="E1:85"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a>  &amp; <font color="Maroon">h1</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a>  )
 ;<br/>

<a NAME="E2:85"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">h1</font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:85"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="group_6.html#K8" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<a NAME="E1:85_1"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:85_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b> <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="group_6.html#K8" target="_self">*</a> <font color="Maroon">h</font></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h1</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:85"/>
<font color="Maroon">h1</font> <a href="group_6.html#K8" target="_self">*</a> <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E5:85"/>
<font color="Maroon">h1</font> <a href="group_6.html#K8" target="_self">*</a> <font color="Maroon">h</font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i>, </i>;<br/>


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