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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">h</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="E1:74"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D12" target="_self">GROUP_6:def 12</a><br/>

<a NAME="E2:74"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">h</font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a> 
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Imh</font> =  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font> as   <a href="group_1.html#NM1">Group</a> ;<br/>
<a NAME="E4:74"/>
<font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Imh</font>,<font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<a NAME="E1:74_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">h</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<font color="Maroon">Imh</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E2:74_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">Imh</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:74_1"/>
<font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Imh</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="group_6.html#T6" target="_self">Th6</a>, <a class="ref" href="funct_2.html#T31">FUNCT_2:31</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Imh</font>,<font color="Maroon">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Imh</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Imh</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><a NAME="E2:74_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Imh</font> &amp; <font color="Maroon">b'</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Imh</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:74_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 <b>by </b><i><a class="txt" href="group_6.html#E1:74"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:74_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="group_6.html#T6" target="_self">Th6</a>, <a class="txt" href="group_6.html#E1:74"><i><font color="Green">E45</font></i></a></i>;<br/><b>thus </b><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> = 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T7" target="_self">Th7</a>, <a class="ref" href="group_6.html#T8" target="_self">Th8</a>, <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon">h'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="group_6.html#T4" target="_self">Th4</a>, <a class="txt" href="group_6.html#E4:74_2"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="group_6.html#T4" target="_self">Th4</a>, <a class="ref" href="group_6.html#T7" target="_self">Th7</a>, <a class="txt" href="group_6.html#E4:74_2"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#T4" target="_self">Th4</a>, <a class="ref" href="group_6.html#T8" target="_self">Th8</a>, <a class="txt" href="group_6.html#E4:74_2"><i><font color="Green">E53</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:74"/>
<font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  is   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


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