<?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> 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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D6">GROUP_2:def 6</a><br/>









<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E4:61_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="group_3.html#E3:61_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<a NAME="E5:61_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="group_3.html#E3:61_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_3.html#T70" target="_self">Th70</a></i>;<br/>
<b>hence </b><a NAME="E6:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>


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

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_2.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:61"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:61"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E6:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_3.html#E2:61"><i><font color="Green">E48</font></i></a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:61"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</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_3.html#T70" target="_self">Th70</a></i>;<br/>
<a NAME="E9:61"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="group_3.html#E5:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="group_3.html#E8:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_3.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E10:61"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="group_3.html#E5:61"><i><font color="Green">E49</font></i></a>, <a class="txt" href="group_3.html#E8:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_2.html#T99">GROUP_2:99</a></i>;<br/>
<b>hence </b><a NAME="E11:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_2.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_3.html#E5:61"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_3.html#T70" target="_self">Th70</a></i>;<br/>


</div>
