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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">x</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="c4">T</font> be   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">E</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">X</font> =  <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="c4">T</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><a NAME="E1:31_1"/>
 <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="c4">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 <b>by </b><i><a class="ref" href="group_10.html#T7" target="_self" title="GROUP_10:th.7">Th7</a></i>;<br/><a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E23">A2</font></i>: 
 ex <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x'</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="c4">T</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>assume </b><a NAME="E3:31_1"/>
 <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="c4">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><a NAME="E4:31_1"/><b>then </b>
not for <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x'</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="c4">T</font> iff <font color="Olive" title="b1">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">x''</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:31_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c6">x''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">x''</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="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E2:31_1"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">y'</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="E8:31_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c6">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y'</font> &amp; <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">y'</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E6:31_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/><a NAME="E9:31_1"/>
 ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon" title="c7">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="group_10.html#E8:31_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="group_10.html#D14" target="_self" title="GROUP_10:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E10:31_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E1:31"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a>, <a class="txt" href="group_10.html#E6:31_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="group_10.html#E8:31_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
 <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="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>


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