<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>, <font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a> <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">A</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">B</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:18"/>
<font color="Maroon">A</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">B</font> ;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A1</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font> <b>by </b><i/>;<br/>
<a NAME="E4:18"/>
for <font color="Olive">b2</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font></span>)</span> holds  <font color="Olive">b2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="group_2.html#K13">*</a> <font color="Olive">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font></span>)</span>;<br/>

<a NAME="E1:18_1"/>
<font color="Maroon">b2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E2:18_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">b2</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">C</font>
 ;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><a NAME="E5:18_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font></span>)</span><b> such that </b><br/><a NAME="E7:18_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/><a NAME="E8:18_1_1"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A1</font><b> such that </b><br/><a NAME="E10:18_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K1" target="_self">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/><b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><a NAME="E12:18_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E13:18_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>then </b><i><font color="Green">E54</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</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">g1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>
;<br/><a NAME="E15:18_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">B</font> ;<br/><font color="Maroon">b1</font> <a href="group_2.html#K12">*</a> <font color="Maroon">A1</font> = 
<font color="Maroon">b</font> <a href="group_2.html#K12">*</a> <font color="Maroon">A1</font>
<b>by </b><i><a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
<br/>.= 
<font color="Maroon">A1</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T140">GROUP_3:140</a></i>
<br/>.= 
<font color="Maroon">A1</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>
;<br/><a NAME="E18:18_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><a NAME="E19:18_1_1"/>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A1</font></span>)</span> <a href="group_2.html#K13">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E20:18_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b2</font>
 <b>by </b><i>, , <a class="ref" href="group_6.html#T2">GROUP_6:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_1"/>
<font color="Maroon">b2</font> <a href="group_2.html#K12">*</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="group_2.html#K13">*</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">A</font> is  <a href="group_1.html#V1">strict</a> <a href="group_3.html#V1">normal</a>  <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">h</font> <a href="grsolv_1.html#K2" target="_self">.:</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="group_3.html#T141">GROUP_3:141</a></i>;<br/>


</div>
