<?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>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>





<a NAME="E1:15"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">y</font> implies  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:15_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:15_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">u</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>
<b>take </b><font color="Maroon">a</font> = <font color="Maroon">u</font>;<br/>

<b>thus </b><a NAME="E6:15_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15"/>
(  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<b>then consider </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><br/><a NAME="E3:15_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E4:15_2"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">h</font> = <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>;<br/>

<b>thus </b><a NAME="E1:15_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T33">GROUP_2:33</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_2.html#T34">GROUP_2:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_2.html#K3">*</a> <font color="Maroon">A</font></span>)</span> <a href="group_2.html#K4">*</a> <font color="Maroon">y</font> iff  ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <b>by </b><i/>;<br/>


</div>
