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<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">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H</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>let </b><font color="Maroon">K</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/>









<a NAME="E1:18"/><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">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">K</font> implies  ex <font color="Olive">g</font>, <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> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T114">GROUP_2:114</a></i>;<br/>
<b>consider </b><font color="Maroon">h1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:18_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<b>thus </b><a NAME="E6:18_1"/>
 ex <font color="Olive">g</font>, <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> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/><i><font color="Green">E42</font></i>: 
(  ex <font color="Olive">g</font>, <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> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> ) implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
 ex <font color="Olive">g</font>, <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> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:18_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> &amp; <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 ;<br/>
<a NAME="E4:18_2"/>
 ex <font color="Olive">g1</font>, <font color="Olive">g2</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">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font> &amp; <font color="Olive">g2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">g2</font> = <font color="Maroon">h</font>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>;<br/>
<a NAME="E1:18_2_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<b>hence </b><a NAME="E2:18_2_1"/>
 ex <font color="Olive">g1</font>, <font color="Olive">g2</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">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font> &amp; <font color="Olive">g2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="group_2.html#T114">GROUP_2:114</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:18"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_2.html#K10">*</a> <font color="Maroon">K</font> iff  ex <font color="Olive">g</font>, <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> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Olive">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">K</font> ) )
 <b>by </b><i>, </i>;<br/>


</div>
