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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">G1</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Gr</font> =  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a1</font></span><a href="domain_1.html#K6">}</a></span> as    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T65">GROUP_2:65</a></i>;<br/>
<a NAME="E3:3"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds <br/> ( <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> iff <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> holds <br/><font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:3_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<a NAME="E4:3_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="group_4.html#T4">GROUP_4:4</a></i>;<br/>
<b>hence </b><a NAME="E5:3_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:3_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font> holds <br/><font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font>
 ;<br/>

<a NAME="E2:3_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:3_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<a NAME="E6:3_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="group_4.html#T4">GROUP_4:4</a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_2"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds <br/> ( <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> iff <font color="Olive">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">Gr</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:3"/>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="group_2.html#T69">GROUP_2:69</a></i>;<br/>


</div>
