<?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">H1</font> be    <a href="group_2.html#M1" target="_self">Subgroup</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>thus </b><a NAME="E1:78"/>
( <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> implies  ex <font color="Olive">H</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">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span>
 ;<br/>

<a NAME="E2:78_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_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="E1:78_1_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"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:78_1_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"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/><b>consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:78_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">a2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:78_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i/>;<br/><i><font color="Green">E53</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="group_2.html#T20" target="_self">Th20</a>, <a class="ref" href="group_2.html#T21" target="_self">Th21</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E8:78_1_1"/>
<font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E10:78_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="group_2.html#T20" target="_self">Th20</a>, <a class="ref" href="group_2.html#T21" target="_self">Th21</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <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="E10:78_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 ;<br/><i><font color="Green">E55</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"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="group_2.html#T22" target="_self">Th22</a>, <a class="ref" href="group_2.html#T26" target="_self">Th26</a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><a NAME="E12:78_1_1"/>
( <font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b2</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T20" target="_self">Th20</a>, <a class="ref" href="group_2.html#T21" target="_self">Th21</a>, <a class="ref" href="group_2.html#T26" target="_self">Th26</a>, <a class="txt" href="group_2.html#E4:78_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E13:78_1_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"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_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="E1:78_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <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="E3:78_1_2"/><i><font color="Green">E57</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">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 ;<br/><b>consider </b><font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:78_1_2"/><i><font color="Green">E58</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">b1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i>, <a class="txt" href="group_2.html#E3:78_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="group_2.html#E5:78_1_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:78_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</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">b1</font> <a href="group_1.html#K3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_2.html#E5:78_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="group_2.html#T20" target="_self">Th20</a>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>;<br/><a NAME="E7:78_1_2"/>
( <font color="Maroon">a1</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b1</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E8:78_1_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"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:78_1"/>
 ex <font color="Olive">H</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">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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

<b>given </b><font color="Maroon">H</font> being    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:78"/><i><font color="Green">E132</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:78"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span>
  <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="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:78_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E3:78_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:78_2"/><b>then </b>
<font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:78_2"/><b>then </b>
<font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <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="E7:78_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 ;<br/>
<i><font color="Green">E135</font></i>: <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</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>
<b>by </b><i>, <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>
;<br/>
<a NAME="E9:78_2"/>
( <font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:78_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</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="E4:78"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <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="E7:78"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_3"/>
not <font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:78_3"/><b>then </b><i><font color="Green">E138</font></i>: 
not <font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 ;<br/><a NAME="E3:78_3"/>
( <font color="Maroon">y</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</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> &amp; <font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">b</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i>, , <a class="ref" href="group_1.html#T25">GROUP_1:25</a></i>;<br/><b>hence </b><a NAME="E4:78_3"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E9:78"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:78"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>
<b>hence </b><a NAME="E11:78"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H1</font></span>)</span> <a href="group_2.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" target="_self">carr</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
