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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>thus </b><a NAME="E1:104"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127" target="_self">Th127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="group_2.html#E1:104_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_2.html#T127" target="_self">Th127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
 <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T132" target="_self">Th132</a></i>
;<br/>
<b>hence </b><a NAME="E3:104_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="group_2.html#T136" target="_self">Th136</a></i>;<br/>


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

<b>assume </b><a NAME="E2:104"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T43" target="_self">Th43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_2.html#T127" target="_self">Th127</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T127" target="_self">Th127</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_2.html#E2:104"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_2.html#T136" target="_self">Th136</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>

;<br/>


</div>
