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<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> 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="E1:103"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:103_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:103_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</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="E3:103_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:103_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</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_2.html#T125" target="_self">Th125</a></i>;<br/>
<a NAME="E5:103_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E1:103_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="group_2.html#E4:103_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a></i>;<br/>
<b>hence </b><a NAME="E6:103_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E3:103_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_2.html#T88" target="_self">Th88</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:103_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:103_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</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_2.html#T88" target="_self">Th88</a></i>;<br/>
<a NAME="E5:103_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49" target="_self">Th49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:103_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E1:103_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_2.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E8:103_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E4:103_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="group_2.html#T59" target="_self">Th59</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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>5</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</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_1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E10:103_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="group_2.html#E8:103_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="group_2.html#T125" target="_self">Th125</a></i>;<br/>


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

<b>assume </b><a NAME="E2:103"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:103"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <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="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</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_2.html#T55" target="_self">Th55</a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E4:103"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="group_2.html#E2:103"><i><font color="Green">E74</font></i></a>, <a class="ref" href="group_2.html#T125" target="_self">Th125</a></i>;<br/>
<b>hence </b><a NAME="E5:103"/>
<font color="Maroon">c<sub>2</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_2.html#T88" target="_self">Th88</a></i>;<br/>


</div>
