<?xml version="1.0"?>
<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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>





<b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>
<a NAME="E1:117"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="group_2.html#T39" target="_self">Th39</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, </i>;<br/>
<a NAME="E4:117"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T52" target="_self">Th52</a>, </i>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:117_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:117_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:117_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:117_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:117_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T34" target="_self">Th34</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="E8:117_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:117_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H2</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E11:117_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E12:117_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E14:117_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>and </b><br/><a NAME="E15:117_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/><i><font color="Green">E131</font></i>: <font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<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"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span>
<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"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span>
<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">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
<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"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E17:117_1"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E18:117_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/><b>hence </b><a NAME="E19:117_1"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>now </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/><b>assume </b><a NAME="E1:117_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:117_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:117_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:117_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font>
 <b>and </b><br/><a NAME="E7:117_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">g2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i>, <a class="txt" href="group_2.html#E32"><i><font color="Green">Lemma64</font></i></a></i>;<br/><i><font color="Green">E136</font></i>: <font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i>, , <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
;<br/><a NAME="E9:117_2"/>
<font color="Maroon">g2</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:117_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">H2</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E11:117_2"/><b>then </b>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="group_2.html#K13" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:117_2"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:117"/>
 ex <font color="Olive">H1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font> st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">H1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_2.html#K12" target="_self">*</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
