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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">b</font> where <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> </span><br/>



<b>set </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:80"/>
<font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green" title="E57">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> </span><br/>

<a NAME="E2:80_1"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="group_3.html#T51" target="_self" title="GROUP_3:th.51">Th51</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="#E1:80_1"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/>
<a NAME="E4:80_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<b>by </b><i><a class="txt" href="#E2:80_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="group_3.html#D7" target="_self" title="GROUP_3:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E5:80_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:80"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:80"/><i><font color="Green" title="E57">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  )
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:80"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="#E4:80"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="group_3.html#D7" target="_self" title="GROUP_3:def.7">Def7</a></i>;<br/>
<b>thus </b><a NAME="E7:80"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K5" title="GROUP_3:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:80"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="#E6:80"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="group_3.html#T51" target="_self" title="GROUP_3:th.51">Th51</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
