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<b>let </b><font color="Maroon">H1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">H2</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>assume </b><b>that </b><br/><a NAME="E1:9_1_2"/><i><font color="Green">E57</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> </span>}</span> 
 <b>and </b><br/><a NAME="E2:9_1_2"/><i><font color="Green">E58</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<a NAME="E3:9_1_2"/>
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds <br/> ( <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font> iff <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1"><b>proof </b></a><div class="add">

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_2_1_1"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:9_1_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<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="E4:9_1_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E5:9_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E6:9_1_2_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:9_1_2_1_1"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1_2_1_2"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:9_1_2_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<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="E4:9_1_2_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E5:9_1_2_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E6:9_1_2_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:9_1_2_1_2"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1_2_1"/>
( <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font> implies <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H1</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9_1_2"/>
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="group_2.html#D6">GROUP_2:def 6</a></i>;<br/>


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