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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>;<br/><b>let </b><font color="Maroon" title="c2">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font>;<br/><b>let </b><font color="Maroon" title="c5">T</font> be   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">E</font>;<br/>







<b>assume </b><a NAME="E1:30"/>
not  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font>
 ;<br/>

<a NAME="E2:30"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:30"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:30"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font> &amp; <font color="Maroon" title="c6">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E4:30"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">z1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><b> such that </b><br/><a NAME="E7:30"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z1'</font> &amp; <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">z1'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">z1''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><b> such that </b><br/><a NAME="E9:30"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c6">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1''</font> &amp; <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c8">z1''</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E5:30"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">z2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><b> such that </b><br/><a NAME="E3:30_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z2'</font> &amp; <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font> )
 ;<br/><a NAME="E4:30_1_1"/>
<font color="Maroon" title="c10">z2'</font>,<font color="Maroon" title="c3">x</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:30_1_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="group_10.html#T5" target="_self" title="GROUP_10:th.5">Th5</a></i>;<br/><a NAME="E5:30_1_1"/><b>then </b>
<font color="Maroon" title="c10">z2'</font>,<font color="Maroon" title="c8">z1''</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:30"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="group_10.html#E9:30"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="group_10.html#T6" target="_self" title="GROUP_10:th.6">Th6</a></i>;<br/><a NAME="E6:30_1_1"/><b>then </b>
<font color="Maroon" title="c8">z1''</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="ref" href="group_10.html#T5" target="_self" title="GROUP_10:th.5">Th5</a></i>;<br/><a NAME="E7:30_1_1"/><b>then </b>
<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E9:30"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="group_10.html#T6" target="_self" title="GROUP_10:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E8:30_1_1"/>
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:30_1_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:30_1"/>
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">z2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">E</font><b> such that </b><br/><a NAME="E4:30_1"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c9">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z2'</font> &amp; <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font> )
 ;<br/><a NAME="E5:30_1"/>
<font color="Maroon" title="c10">z2'</font>,<font color="Maroon" title="c4">y</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:30_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="group_10.html#T5" target="_self" title="GROUP_10:th.5">Th5</a></i>;<br/><a NAME="E6:30_1"/><b>then </b>
<font color="Maroon" title="c10">z2'</font>,<font color="Maroon" title="c7">z1'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:30"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="group_10.html#E9:30"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="group_10.html#T6" target="_self" title="GROUP_10:th.6">Th6</a></i>;<br/><a NAME="E7:30_1"/><b>then </b>
<font color="Maroon" title="c7">z1'</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="ref" href="group_10.html#T5" target="_self" title="GROUP_10:th.5">Th5</a></i>;<br/><a NAME="E8:30_1"/><b>then </b>
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">z2'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:30"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="group_10.html#T6" target="_self" title="GROUP_10:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E9:30_1"/>
<font color="Maroon" title="c9">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:30_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:30"/>
 <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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