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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>;<br/><b>let </b><font color="Maroon" title="c3">h</font> be   <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>;<br/>





<b>thus </b><a NAME="E1:81"/>
( <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a>  implies (  <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> &amp; <font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/>
( <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D14" target="_self" title="GROUP_6:def.14">GROUP_6:def 14</a><br/>

<b>hence </b><a NAME="E2:81_1"/>
(  <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> &amp; <font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="group_6.html#D12" target="_self" title="GROUP_6:def.12">Def12</a>, <a class="ref" href="group_6.html#D13" target="_self" title="GROUP_6:def.13">Def13</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:81"/>
(  <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> &amp; <font color="Maroon" title="c3">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 ;<br/>

<b>hence </b><a NAME="E3:81"/>
( <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  &amp; <font color="Maroon" title="c3">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  )
 <b>by </b><i><a class="ref" href="group_6.html#D12" target="_self" title="GROUP_6:def.12">Def12</a>, <a class="ref" href="group_6.html#D13" target="_self" title="GROUP_6:def.13">Def13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D14" target="_self" title="GROUP_6:def.14">GROUP_6:def 14</a><br/>


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