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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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>







<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font> implies  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:61_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:61_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E7:61_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="group_6.html#E7:61_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E9:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_6.html#T2" target="_self">Th2</a></i>;<br/>


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

<b>given </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:61"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E3:61"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:61"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:61"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:61"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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