<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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="c2">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<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="c2">G</font>,<font color="Maroon" title="c1">E</font>;<br/>







<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">O1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">O2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">O</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> ;<br/>
<div><i><font color="Green" title="E27">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</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="E1:36_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">X</font> &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Maroon" title="c9">X</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="Olive" title="b1">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/><a NAME="E4:36_1_1"/>
not <font color="Maroon" title="c9">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="group_10.html#T7" target="_self" title="GROUP_10:th.7">Th7</a></i>;<br/><a NAME="E5:36_1_1"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><a NAME="E6:36_1_1"/><b>then </b>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font>
 ;<br/><a NAME="E7:36_1_1"/><b>then </b>
<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:36_1_1"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font> )
 </span><b>by </b><i><a class="txt" href="group_10.html#E8:36_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c9">X</font>
 ;<br/><div class="add"><a NAME="E2:36_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:36_1_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>assume </b><a NAME="E2:36_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_2"/>
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="group_10.html#E2:36_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1"><b>suppose </b></a><a NAME="E1:36_1_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:36_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/><b>hence </b><a NAME="E3:36_1_2_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_2"><b>suppose </b></a><a NAME="E1:36_1_2_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:36_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/><b>hence </b><a NAME="E3:36_1_2_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:36"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">O</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="group_10.html#E2:36"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E5:36"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:36"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:36"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">f1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E8:36"/><i><font color="Green" title="E29">A7</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  &amp; <font color="Maroon" title="c9">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<a NAME="E9:36"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c4">T</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:36"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:36"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">f2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E12:36"/><i><font color="Green" title="E30">A8</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  &amp; <font color="Maroon" title="c10">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T73" title="FINSEQ_4:th.73">FINSEQ_4:73</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c9">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">f2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:36_2"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 ;<br/><a NAME="E3:36_2"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="group_10.html#E2:36_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:36_2"/><b>then </b>
(  ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x'</font>,<font color="Maroon" title="c4">T</font> ) &amp;  ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x'</font>,<font color="Maroon" title="c4">T</font> ) )
 ;<br/><b>hence </b><a NAME="E5:36_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E14:36"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<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="E15:36"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">f2</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="group_10.html#E12:36"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E16:36"/><b>then </b><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c9">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="group_10.html#E12:36"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/>
<a NAME="E17:36"/><i><font color="Green" title="E32">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">O</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="group_10.html#E12:36"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c9">f1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c10">f2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c8">O</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c13">p1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E20:36"/><i><font color="Green" title="E33">A12</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1</font></span>)</span> holds <br/><font color="Maroon" title="c13">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E21:36"/><i><font color="Green" title="E34">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:36"/>
for <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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">p1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_3"/><i><font color="Green" title="E35">A14</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">p1</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:36_3"/><i><font color="Green" title="E36">A15</font></i>: 
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">p1</font> &amp; <font color="Maroon" title="c13">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E1:36_3"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:36_3"/><i><font color="Green" title="E37">A16</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E21:36"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="group_10.html#E3:36_3"><i><font color="Green" title="E36">A15</font></i></a></i>;<br/>
<a NAME="E5:36_3"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E3:36_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:36_3"/><b>then </b>
<font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:36_3"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> )
 
;<br/>
<b>hence </b><a NAME="E8:36_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E4:36_3"><i><font color="Green" title="E37">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:36"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">c1</font> = <font color="Maroon" title="c13">p1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E25:36"/><i><font color="Green" title="E35">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/>
<a NAME="E26:36"/><i><font color="Green" title="E36">A18</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font> holds <br/><font color="Maroon" title="c14">c1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_4"/><i><font color="Green" title="E37">A19</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font>
 ;<br/>

<a NAME="E2:36_4"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E1:36_4"><i><font color="Green" title="E37">A19</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:36_4"/><b>then </b>
<font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:36_4"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> )
 
;<br/>
<b>hence </b><a NAME="E5:36_4"/>
<font color="Maroon" title="c14">c1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E21:36"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="group_10.html#E1:36_4"><i><font color="Green" title="E37">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:36"/>
for <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="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_5"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font>
 ;<br/>

<a NAME="E2:36_5"/><b>then </b>
 ex <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E3:36_5"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="group_10.html#E26:36"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:36_5"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:36"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E29:36"/><i><font color="Green" title="E38">A21</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_6_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_1"><b>suppose </b></a><a NAME="E1:36_6_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_6_1_1"/>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_6_1_2"><b>suppose </b></a><a NAME="E1:36_6_1_2"/><i><font color="Green" title="E39">A22</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><a NAME="E2:36_6_1_2"/>
for <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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_6_1_2_1"/><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E2:36_6_1_2_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:36_6_1_2_1"/><b>then </b><i><font color="Green" title="E41">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:36_6_1_2"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:36_6_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">c1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="group_10.html#E26:36"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>
<a NAME="E5:36_6_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">c1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:36_6_1_2_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:36_6_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_6_1_2_1"><i><font color="Green" title="E41">A24</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:36_6_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:36_6_1_2"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">c1</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">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E28:36"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E5:36_6_1_2"/><b>then </b>
<font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1
 <b>by </b><i><a class="ref" href="funcop_1.html#T15" title="FUNCOP_1:th.15">FUNCOP_1:15</a></i>;<br/><a NAME="E6:36_6_1_2"/><b>then </b>
<font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:36_6_1_2"/><b>then </b>
<font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1
 <b>by </b><i><a class="ref" href="finseq_2.html#D2" title="FINSEQ_2:def.2">FINSEQ_2:def 2</a></i>;<br/><a NAME="E8:36_6_1_2"/><b>then </b>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
 <b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>;<br/><b>hence </b><a NAME="E9:36_6_1_2"/>
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">X</font> &amp; <font color="Olive" title="b2">X</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="Olive" title="b1">x</font>,<font color="Maroon" title="c4">T</font> &amp; <font color="Olive" title="b1">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/>
<a NAME="E30:36"/><i><font color="Green" title="E39">A25</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</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#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font>] holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
;<br/>
<a NAME="E31:36"/><i><font color="Green" title="E40">A26</font></i>: 
for <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#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_7"/><i><font color="Green" title="E41">A27</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">y</font> =  <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c16">x'</font>,<font color="Maroon" title="c4">T</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c17">y</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_7_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon" title="c18">x</font> = <font color="Maroon" title="c16">x'</font>;<br/><b>set </b><font color="Maroon" title="c19">X</font> = <font color="Maroon" title="c17">y</font>;<br/><b>take </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c16">x'</font>;<br/><b>take </b><font color="Maroon" title="c21">X</font> = <font color="Maroon" title="c17">y</font>;<br/><b>thus </b><a NAME="E1:36_7_1"/>
( <font color="Maroon" title="c16">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">X</font> &amp; <font color="Maroon" title="c21">X</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="c20">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/><a NAME="E2:36_7_1"/>
 <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x''</font> where <font color="Olive" title="b1">x''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : <font color="Olive" title="b1">x''</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="group_10.html#D13" target="_self" title="GROUP_10:def.13">Def13</a></i>;<br/><a NAME="E3:36_7_1"/><b>then </b>
 ex <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b1">e</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E1:36_7"><i><font color="Green" title="E41">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:36_7_1"/>
<font color="Maroon" title="c20">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/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:36_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c17">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">FX</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E33:36"/><i><font color="Green" title="E41">A28</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">FX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font> &amp; ( for <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#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="group_10.html#E30:36"><i><font color="Green" title="E39">A25</font></i></a>, <a class="txt" href="group_10.html#E31:36"><i><font color="Green" title="E40">A26</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">y</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="E1:36_8_1"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c17">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E3:36_8_1"/><i><font color="Green" title="E42">A29</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Maroon" title="c17">X</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="Olive" title="b1">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c18">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E5:36_8_1"/><i><font color="Green" title="E43">A30</font></i>: 
<font color="Maroon" title="c17">X</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="c18">x'</font>,<font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_8_1"><i><font color="Green" title="E42">A29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x'</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font>;<br/><a NAME="E7:36_8_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_8_1"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">x''</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:36_8_1"/><i><font color="Green" title="E44">A31</font></i>: 
<font color="Maroon" title="c17">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">x''</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_8_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c22">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><a NAME="E2:36_8_1_1"/><i><font color="Green" title="E45">A32</font></i>: 
<font color="Maroon" title="c18">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="Maroon" title="c22">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x'</font>
 ;<br/><b>set </b><font color="Maroon" title="c23">y'</font> = <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="Maroon" title="c22">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x'</font>;<br/><a NAME="E3:36_8_1_1"/>
<font color="Maroon" title="c18">x'</font>,<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="Maroon" title="c22">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x'</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="ref" href="group_10.html#D14" target="_self" title="GROUP_10:def.14">Def14</a></i>;<br/><a NAME="E4:36_8_1_1"/><b>then </b>
<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="Maroon" title="c22">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y''</font> where <font color="Olive" title="b1">y''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : <font color="Maroon" title="c18">x'</font>,<font color="Olive" title="b1">y''</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c4">T</font> </span>}</span> 
 ;<br/><a NAME="E5:36_8_1_1"/><b>then </b><i><font color="Green" title="E46">A33</font></i>: 
<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="Maroon" title="c22">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x''</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:36_8_1"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="group_10.html#E9:36_8_1"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:36_8_1_1"/>
<font color="Maroon" title="c18">x'</font>,<font color="Maroon" title="c18">x'</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="ref" href="group_10.html#T4" target="_self" title="GROUP_10:th.4">Th4</a></i>;<br/><a NAME="E7:36_8_1_1"/><b>then </b>
<font color="Maroon" title="c18">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y''</font> where <font color="Olive" title="b1">y''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : <font color="Maroon" title="c18">x'</font>,<font color="Olive" title="b1">y''</font> <a href="group_10.html#R2" title="GROUP_10:pred.2">are_conjugated_under</a> <font color="Maroon" title="c4">T</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E8:36_8_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E5:36_8_1"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="group_10.html#E9:36_8_1"><i><font color="Green" title="E44">A31</font></i></a>, <a class="txt" href="group_10.html#E2:36_8_1_1"><i><font color="Green" title="E45">A32</font></i></a>, <a class="txt" href="group_10.html#E5:36_8_1_1"><i><font color="Green" title="E46">A33</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E11:36_8_1"/><b>then </b>
<font color="Maroon" title="c18">x'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="ref" href="group_10.html#D12" target="_self" title="GROUP_10:def.12">Def12</a></i>;<br/><a NAME="E12:36_8_1"/><b>then </b>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x'''</font> where <font color="Olive" title="b1">x'''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> : <font color="Olive" title="b1">x'''</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> </span>}</span> 
 ;<br/><a NAME="E13:36_8_1"/><b>then </b><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="ref" href="group_10.html#D13" target="_self" title="GROUP_10:def.13">Def13</a></i>;<br/><a NAME="E14:36_8_1"/><b>then </b>
 ex <font color="Olive" title="b1">x''</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> ex <font color="Olive" title="b2">X'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">x''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Olive" title="b2">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> &amp; <font color="Olive" title="b2">X'</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="Olive" title="b1">x''</font>,<font color="Maroon" title="c4">T</font> &amp; <font color="Olive" title="b1">x''</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><b>hence </b><a NAME="E15:36_8_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">FX</font>
 <b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a>, <a class="txt" href="group_10.html#E3:36_8_1"><i><font color="Green" title="E42">A29</font></i></a>, <a class="txt" href="group_10.html#E5:36_8_1"><i><font color="Green" title="E43">A30</font></i></a>, <a class="txt" href="group_10.html#E13:36_8_1"><i><font color="Green" title="E45">A34</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:36_8"/><i><font color="Green" title="E42">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c16">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">FX</font>
 ;<br/><a NAME="E4:36_8"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">FX</font> &amp; <font color="Maroon" title="c16">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_8"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c19">X'</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E6:36_8"/><i><font color="Green" title="E43">A36</font></i>: 
( <font color="Maroon" title="c18">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c19">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y</font> &amp; <font color="Maroon" title="c19">X'</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="c18">x'</font>,<font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c18">x'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><a NAME="E7:36_8"/>
<font color="Maroon" title="c19">X'</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="c18">x'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E6:36_8"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="group_10.html#T9" target="_self" title="GROUP_10:th.9">Th9</a></i>;<br/><a NAME="E8:36_8"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c19">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/><b>hence </b><a NAME="E9:36_8"/>
<font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="group_10.html#E6:36_8"><i><font color="Green" title="E43">A36</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E35:36"/><b>then </b><i><font color="Green" title="E42">A37</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">FX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:36_9"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">FX</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c18">x1'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c19">X1</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E3:36_9"/><i><font color="Green" title="E43">A38</font></i>: 
( <font color="Maroon" title="c18">x1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c19">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c19">X1</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="c18">x1'</font>,<font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c18">x1'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><b>assume </b><a NAME="E4:36_9"/>
<font color="Maroon" title="c17">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">FX</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c20">x2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c21">X2</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E6:36_9"/><i><font color="Green" title="E44">A39</font></i>: 
( <font color="Maroon" title="c20">x2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font> &amp; <font color="Maroon" title="c21">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font> &amp; <font color="Maroon" title="c21">X2</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="c20">x2'</font>,<font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c20">x2'</font> <a href="group_10.html#R1" title="GROUP_10:pred.1">is_fixed_under</a> <font color="Maroon" title="c4">T</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a></i>;<br/><b>assume </b><a NAME="E7:36_9"/>
<font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">FX</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x2</font>
 ;<br/><b>then </b><span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c18">x1'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> = 
 <a href="group_10.html#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c20">x2'</font>,<font color="Maroon" title="c4">T</font>
<b>by </b><i><a class="txt" href="group_10.html#E3:36_9"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="group_10.html#E6:36_9"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="group_10.html#T9" target="_self" title="GROUP_10:th.9">Th9</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">x2'</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="group_10.html#E6:36_9"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="group_10.html#T9" target="_self" title="GROUP_10:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E9:36_9"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:36_9"><i><font color="Green" title="E43">A38</font></i></a>, <a class="txt" href="group_10.html#E6:36_9"><i><font color="Green" title="E44">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E37:36"/><b>then </b>
<font color="Maroon" title="c15">FX</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E38:36"/><b>then </b><i><font color="Green" title="E43">A40</font></i>: 
 <a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font>,<span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="group_10.html#E33:36"><i><font color="Green" title="E41">A28</font></i></a>, <a class="txt" href="group_10.html#E35:36"><i><font color="Green" title="E42">A37</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c16">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E40:36"/><i><font color="Green" title="E44">A41</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font></span>)</span> holds <br/><font color="Maroon" title="c16">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E41:36"/><i><font color="Green" title="E45">A42</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E42:36"/>
for <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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p2</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:36_10"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:36_10"/><i><font color="Green" title="E46">A43</font></i>: 
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p2</font> &amp; <font color="Maroon" title="c16">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:36_10"/><i><font color="Green" title="E47">A44</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E41:36"><i><font color="Green" title="E45">A42</font></i></a>, <a class="txt" href="group_10.html#E3:36_10"><i><font color="Green" title="E46">A43</font></i></a></i>;<br/>
<a NAME="E5:36_10"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E3:36_10"><i><font color="Green" title="E46">A43</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E6:36_10"/><b>then </b>
<font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E12:36"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:36_10"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> )
 
;<br/>
<b>hence </b><a NAME="E8:36_10"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E4:36_10"><i><font color="Green" title="E47">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E43:36"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">c2</font> = <font color="Maroon" title="c16">p2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c14">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">c2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">c</font> = <font color="Maroon" title="c14">c1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">c2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E46:36"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E47:36"/><b>then </b><i><font color="Green" title="E46">A45</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E48:36"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c</font> holds <br/><font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_11"/><i><font color="Green" title="E47">A46</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_11_1_1"/>
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font> or  ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c2</font> &amp; <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> ) )
 </span><b>by </b><i><a class="txt" href="group_10.html#E1:36_11"><i><font color="Green" title="E47">A46</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_11_1_1_1"><b>suppose </b></a><a NAME="E1:36_11_1_1_1"/><i><font color="Green" title="E48">A47</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">c1</font>
 ;<br/><div class="add"><a NAME="E2:36_11_1_1_1"/><b>then </b><i><font color="Green" title="E49">A48</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c14">c1</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="group_10.html#E1:36_11_1_1_1"><i><font color="Green" title="E48">A47</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E21:36"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="group_10.html#E1:36_11_1_1_1"><i><font color="Green" title="E48">A47</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E2:36_11_1_1_1"><i><font color="Green" title="E49">A48</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E4:36_11_1_1_1"/>
<font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_11_1_1_2"><b>suppose </b></a><a NAME="E1:36_11_1_1_2"/>
 ex <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c2</font> &amp; <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">j</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c21">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:36_11_1_1_2"/><i><font color="Green" title="E48">A49</font></i>: 
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c2</font>
 <b>and </b><br/><a NAME="E4:36_11_1_1_2"/><i><font color="Green" title="E49">A50</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c21">j</font>
 ;<br/><a NAME="E5:36_11_1_1_2"/><i><font color="Green" title="E50">A51</font></i>: 
<font color="Maroon" title="c21">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E3:36_11_1_1_2"><i><font color="Green" title="E48">A49</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c17">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font>
<b>by </b><i><a class="txt" href="group_10.html#E3:36_11_1_1_2"><i><font color="Green" title="E48">A49</font></i></a>, <a class="txt" href="group_10.html#E4:36_11_1_1_2"><i><font color="Green" title="E49">A50</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">j</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E41:36"><i><font color="Green" title="E45">A42</font></i></a>, <a class="txt" href="group_10.html#E3:36_11_1_1_2"><i><font color="Green" title="E48">A49</font></i></a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E20:36"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="group_10.html#E4:36_11_1_1_2"><i><font color="Green" title="E49">A50</font></i></a>, <a class="txt" href="group_10.html#E5:36_11_1_1_2"><i><font color="Green" title="E50">A51</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
;<br/><b>hence </b><a NAME="E7:36_11_1_1_2"/>
<font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_11"/>
<font color="Maroon" title="c19">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green" title="E47">A52</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">E</font> = 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c19">c</font>
<b>by </b><i><a class="txt" href="group_10.html#E16:36"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="group_10.html#E17:36"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="group_10.html#E47:36"><i><font color="Green" title="E46">A45</font></i></a>, <a class="ref" href="weddwitt.html#T6" title="WEDDWITT:th.6">WEDDWITT:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c17">c2</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c20">p'</font> = <font color="Maroon" title="c3">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E51:36"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c2</font> holds <br/><font color="Maroon" title="c20">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:36_13"/><i><font color="Green" title="E48">A53</font></i>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">c2</font>
 ;<br/>

<a NAME="E2:36_13"/>
<font color="Maroon" title="c17">c2</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E41:36"><i><font color="Green" title="E45">A42</font></i></a>, <a class="txt" href="group_10.html#E1:36_13"><i><font color="Green" title="E48">A53</font></i></a></i>;<br/>
<a NAME="E3:36_13"/><b>then </b><i><font color="Green" title="E49">A54</font></i>: 
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:36_13"><i><font color="Green" title="E48">A53</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E4:36_13"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E40:36"><i><font color="Green" title="E44">A41</font></i></a>, <a class="txt" href="group_10.html#E1:36_13"><i><font color="Green" title="E48">A53</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:36_13"/><b>then </b>
<font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">X</font> where <font color="Olive" title="b1">X</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> : (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Olive" title="b1">X</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="Olive" title="b2">x</font>,<font color="Maroon" title="c4">T</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_10.html#E12:36"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E7:36_13"/><i><font color="Green" title="E50">A55</font></i>: 
( <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">X</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c22">X</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> st <font color="Maroon" title="c22">X</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="Olive" title="b1">x</font>,<font color="Maroon" title="c4">T</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c23">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E9:36_13"/><i><font color="Green" title="E51">A56</font></i>: 
<font color="Maroon" title="c22">X</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="c23">x</font>,<font color="Maroon" title="c4">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:36_13"><i><font color="Green" title="E50">A55</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">H</font> =  <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font>;<br/>
<a NAME="E10:36_13"/><i><font color="Green" title="E52">A57</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</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="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_10.html#T10" target="_self" title="GROUP_10:th.10">Th10</a></i>;<br/>
<a NAME="E11:36_13"/><i><font color="Green" title="E53">A58</font></i>: 
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:36_13"><i><font color="Green" title="E49">A54</font></i></a>, <a class="txt" href="group_10.html#E7:36_13"><i><font color="Green" title="E50">A55</font></i></a>, <a class="txt" href="group_10.html#E9:36_13"><i><font color="Green" title="E51">A56</font></i></a>, <a class="ref" href="group_10.html#T10" target="_self" title="GROUP_10:th.10">Th10</a></i>;<br/>
<a NAME="E12:36_13"/><i><font color="Green" title="E54">A59</font></i>: 
 ex <font color="Olive" title="b1">B</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span> &amp; <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_13_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c25">B</font> =  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="group_2.html#T164" title="GROUP_2:th.164">GROUP_2:164</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c25">B</font>
;<br/>

<b>thus </b><a NAME="E2:36_13_1"/>
<font color="Maroon" title="c25">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E3:36_13_1"/>
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c25">B</font>
 <b>by </b><i><a class="txt" href="group_10.html#E11:36_13"><i><font color="Green" title="E53">A58</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:36_13"/><b>then </b>
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></i>;<br/>
<a NAME="E14:36_13"/><b>then </b><i><font color="Green" title="E55">A60</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p2">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c25">r</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E16:36_13"/><i><font color="Green" title="E56">A61</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c25">r</font>
 <b>by </b><i><a class="txt" href="group_10.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<a NAME="E17:36_13"/><i><font color="Green" title="E57">A62</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<div><i><font color="Green" title="E58">A63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:36_13_2"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font>
 ;<br/><a NAME="E2:36_13_2"/><b>then </b>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p2">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/><a NAME="E3:36_13_2"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p3">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E17:36_13"><i><font color="Green" title="E57">A62</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><a NAME="E4:36_13_2"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p2">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>;<br/><a NAME="E5:36_13_2"/><b>then </b><i><font color="Green" title="E59">A64</font></i>: 
1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <span class="p2">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="group_10.html#E17:36_13"><i><font color="Green" title="E57">A62</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><b>consider </b><font color="Maroon" title="c26">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:36_13_2"/><i><font color="Green" title="E60">A65</font></i>: 
( <font color="Maroon" title="c26">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <span class="p1">(<span class="default"><a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c23">x</font>,<font color="Maroon" title="c4">T</font></span>)</span> &amp; <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c26">B</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E12:36_13"><i><font color="Green" title="E54">A59</font></i></a></i>;<br/><b>thus </b><a NAME="E8:36_13_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E7:36_13"><i><font color="Green" title="E50">A55</font></i></a>, <a class="txt" href="group_10.html#E9:36_13"><i><font color="Green" title="E51">A56</font></i></a>, <a class="txt" href="group_10.html#E10:36_13"><i><font color="Green" title="E52">A57</font></i></a>, <a class="txt" href="group_10.html#E5:36_13_2"><i><font color="Green" title="E59">A64</font></i></a>, <a class="txt" href="group_10.html#E7:36_13_2"><i><font color="Green" title="E60">A65</font></i></a>, <a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c26">i'</font> = <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">r'</font> = <font color="Maroon" title="c25">r</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E20:36_13"/><i><font color="Green" title="E59">A66</font></i>: 
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c20">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c27">r'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E14:36_13"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="group_10.html#E16:36_13"><i><font color="Green" title="E56">A61</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_13_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_13_3"/>
( <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Olive" title="b1">k</font> )
 </span><b>by </b><i><a class="txt" href="group_10.html#E20:36_13"><i><font color="Green" title="E59">A66</font></i></a>, <a class="ref" href="pepin.html#T33" title="PEPIN:th.33">PEPIN:33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_13_3_1"><b>suppose </b></a><a NAME="E1:36_13_3_1"/>
<font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_13_3_1"/>
<font color="Maroon" title="c20">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="txt" href="group_10.html#E14:36_13"><i><font color="Green" title="E55">A60</font></i></a>, <a class="txt" href="group_10.html#E18:36_13"><i><font color="Green" title="E58">A63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_13_3_2"><b>suppose </b></a><a NAME="E1:36_13_3_2"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Olive" title="b1">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_13_3_2"/>
<font color="Maroon" title="c20">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c17">c2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c21">i</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E52:36"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c17">c2</font>
 
<b>by </b><i><a class="ref" href="weddwitt.html#T5" title="WEDDWITT:th.5">WEDDWITT:5</a></i>;<br/>
<a NAME="E53:36"/><b>then </b>
 ex <font color="Olive" title="b1">t</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c17">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">t</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E54:36"/><b>then </b><i><font color="Green" title="E48">A67</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c17">c2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>;<br/>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c17">c2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="group_10.html#E49:36"><i><font color="Green" title="E47">A52</font></i></a>, <a class="ref" href="nat_d.html#T23" title="NAT_D:th.23">NAT_D:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c14">c1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="txt" href="group_10.html#E54:36"><i><font color="Green" title="E48">A67</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E56:36"/>
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="group_10.html#K12" title="GROUP_10:func.12">the_fixed_points_of</a> <font color="Maroon" title="c4">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E8:36"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="group_10.html#E25:36"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="group_10.html#E29:36"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="group_10.html#E38:36"><i><font color="Green" title="E43">A40</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>


</div>
