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<div class="add">

<b>let </b><font color="Maroon">G1</font> be   <a href="group_1.html#NM1">Group</a>, <font color="Maroon">G2</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/><b>let </b><font color="Maroon">H1</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font>;<br/>





<a NAME="E1:6"/><i><font color="Green">E23</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:6"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<a NAME="E3:6"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> ;<br/>
<a NAME="E5:6"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:6_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:6_1"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:6_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:6_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:6_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:6"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:6_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:6_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E6:6_2"/><i><font color="Green">E42</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">H1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:6_2"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:6_2"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:6_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <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">H1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:6_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:6_2"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , <a class="ref" href="latsubgr.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">H2</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E9:6"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">H2</font>
;<br/>

<b>thus </b><a NAME="E10:6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i/>;<br/>


</div>
