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<b>let </b><font color="Maroon" title="c1">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="c2">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>reconsider </b><font color="Maroon" title="c3">p'</font> = <font color="Maroon" title="c2">p</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <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="E2:45"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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/>
<b>then consider </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">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="E4:45"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font> &amp; not <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E31"><i><font color="Green" title="E27">Lm1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">n</font> =  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">k1</font> = <font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">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/>
<b>reconsider </b><font color="Maroon" title="c8">x1</font> = 1, <font color="Maroon" title="c9">x2</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E7:45"/><i><font color="Green" title="E37">A3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <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="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E8:45"/>
<font color="Maroon" title="c7">k1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E9:45"/><b>then </b>
<font color="Maroon" title="c7">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E2:45"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E10:45"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">k1</font> <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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E7:45"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<a NAME="E11:45"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">k1</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">SF</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c7">k1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E12:45"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c7">k1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span> <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="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c7">k1</font>,<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_10.html#T3" target="_self" title="GROUP_10:th.3">Th3</a></i>;<br/>
<a NAME="E13:45"/><b>then </b><i><font color="Green" title="E39">A5</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c7">k1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span> <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="c1">G</font></span>)</span> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c7">k1</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E7:45"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="card_fin.html#T18" title="CARD_FIN:th.18">CARD_FIN:18</a></i>;<br/>
<a NAME="E14:45"/><b>then </b><i><font color="Green" title="E40">A6</font></i>: 
<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#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c7">k1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</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="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_10.html#E35"><i><font color="Green" title="E31">Lm5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">LO</font> =  <a href="group_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>set </b><font color="Maroon" title="c12">LO'</font> =  <a href="group_10.html#K5" title="GROUP_10:func.5">the_extension_of_left_operation_of</a> <font color="Maroon" title="c7">k1</font>,<span class="p1">(<span class="default"><a href="group_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">E</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c7">k1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> as  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>  <b>by </b><i><a class="txt" href="group_10.html#E14:45"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">T</font> =  <a href="group_10.html#K5" title="GROUP_10:func.5">the_extension_of_left_operation_of</a> <font color="Maroon" title="c7">k1</font>,<span class="p1">(<span class="default"><a href="group_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font></span>)</span> as   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c13">E</font> ;<br/>
<a NAME="E17:45"/><i><font color="Green" title="E41">A7</font></i>: 
 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="c13">E</font> st <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#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E42">A8</font></i>: 
for <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="c13">E</font> holds  <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#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">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> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">f</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E43">A9</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font> <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="c14">T</font> &amp; <font color="Maroon" title="c15">f</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>reconsider </b><font color="Maroon" title="c16">f</font> = <font color="Maroon" title="c15">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="group_10.html#K14" title="GROUP_10:func.14">the_orbits_of</a> <font color="Maroon" title="c14">T</font> <b>by </b><i><a class="txt" href="group_10.html#E3:45_1"><i><font color="Green" title="E43">A9</font></i></a>, <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="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c17">pf</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E6:45_1"/><i><font color="Green" title="E44">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">pf</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="c16">f</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="c16">f</font></span>)</span> holds <br/><font color="Maroon" title="c17">pf</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="E7:45_1"/><i><font color="Green" title="E45">A11</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">pf</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="c16">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</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="E8:45_1"/><i><font color="Green" title="E46">A12</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#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="c16">f</font></span>)</span> holds <br/><font color="Maroon" title="c17">pf</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>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a></i>;<br/>
<a NAME="E9:45_1"/>
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="c17">pf</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="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E47">A13</font></i>: 
<font color="Maroon" title="c18">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="c17">pf</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c19">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green" title="E48">A14</font></i>: 
( <font color="Maroon" title="c19">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">pf</font> &amp; <font color="Maroon" title="c17">pf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E1:45_1_1"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E4:45_1_1"/>
<font color="Maroon" title="c19">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">f</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:45_1_1"/><b>then </b>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</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="c14">T</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:45_1"><i><font color="Green" title="E43">A9</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="c20">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c13">E</font><b> such that </b><br/><a NAME="E7:45_1_1"/><i><font color="Green" title="E49">A15</font></i>: 
( <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">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="c13">E</font> st <font color="Maroon" title="c20">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="c14">T</font> )
 ;<br/>
<a NAME="E8:45_1_1"/>
<font color="Maroon" title="c18">x</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="c20">X</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="group_10.html#E7:45_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_1"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="group_10.html#E7:45_1_1"><i><font color="Green" title="E49">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:45_1_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:45_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">pf</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="c18">c</font> = <font color="Maroon" title="c17">pf</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="E12:45_1"/><i><font color="Green" title="E47">A16</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c18">c</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:45_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="group_10.html#E7:45_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="group_10.html#E8:45_1"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="weddwitt.html#T6" title="WEDDWITT:th.6">WEDDWITT:6</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>  = <span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c2">p</font>;<br/>
<b>consider </b><font color="Maroon" title="c19">c'</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E14:45_1"/><i><font color="Green" title="E48">A17</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="c18">c</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="c18">c</font></span>)</span> holds <br/><font color="Maroon" title="c19">c'</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="E15:45_1"/>
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="c19">c'</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="45_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:45_1_2"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c20">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="c19">c'</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c21">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_1_2"/><i><font color="Green" title="E50">A19</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="c19">c'</font> &amp; <font color="Maroon" title="c19">c'</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="c20">x</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E1:45_1_2"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">i</font> = <font color="Maroon" title="c21">i</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="E5:45_1_2"/><i><font color="Green" title="E51">A20</font></i>: 
<font color="Maroon" title="c22">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="c18">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E52">A21</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">pf</font> = 
 <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="c18">c</font></span>)</span>
<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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">c'</font>
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1"><i><font color="Green" title="E48">A17</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="E7:45_1_2"/>
<font color="Maroon" title="c22">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">f</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="group_10.html#E6:45_1_2"><i><font color="Green" title="E52">A21</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E8:45_1_2"/><b>then </b>
<font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</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="c14">T</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:45_1"><i><font color="Green" title="E43">A9</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="c23">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c13">E</font><b> such that </b><br/><a NAME="E10:45_1_2"/><i><font color="Green" title="E53">A22</font></i>: 
( <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">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="c13">E</font> st <font color="Maroon" title="c23">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="c14">T</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c24">x'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">E</font><b> such that </b><br/><a NAME="E12:45_1_2"/><i><font color="Green" title="E54">A23</font></i>: 
<font color="Maroon" title="c23">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="c24">x'</font>,<font color="Maroon" title="c14">T</font>
 <b>by </b><i><a class="txt" href="group_10.html#E10:45_1_2"><i><font color="Green" title="E53">A22</font></i></a></i>;<br/>
<a NAME="E13:45_1_2"/><i><font color="Green" title="E55">A24</font></i>: 
<font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">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="c23">X</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E6:45_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="txt" href="group_10.html#E7:45_1"><i><font color="Green" title="E45">A11</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="group_10.html#E6:45_1_2"><i><font color="Green" title="E52">A21</font></i></a>, <a class="txt" href="group_10.html#E10:45_1_2"><i><font color="Green" title="E53">A22</font></i></a></i>;<br/>
<a NAME="E14:45_1_2"/><i><font color="Green" title="E56">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">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="txt" href="group_10.html#E1:45_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="group_10.html#E12:45_1_2"><i><font color="Green" title="E54">A23</font></i></a>, <a class="txt" href="group_10.html#E13:45_1_2"><i><font color="Green" title="E55">A24</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p'</font>;<br/>
<font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p'</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</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>
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">p'</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c3">p'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1_2"><i><font color="Green" title="E56">A25</font></i></a></i>
;<br/>
<b>then consider </b><font color="Maroon" title="c26">q</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="E17:45_1_2"/><i><font color="Green" title="E57">A26</font></i>: 
<font color="Maroon" title="c18">c</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c26">q</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c27">p''</font> = <font color="Maroon" title="c3">p'</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<font color="Maroon" title="c20">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">p''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c26">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c27">p''</font>
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="group_10.html#E3:45_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="txt" href="group_10.html#E5:45_1_2"><i><font color="Green" title="E51">A20</font></i></a>, <a class="txt" href="group_10.html#E17:45_1_2"><i><font color="Green" title="E57">A26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">p''</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c27">p''</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c26">q</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<b>hence </b><a NAME="E20:45_1_2"/>
<font color="Maroon" title="c20">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="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:45_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">c'</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="c20">c'</font> = <font color="Maroon" title="c19">c'</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/>
<i><font color="Green" title="E49">A27</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">c</font> = 
 <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="c20">c'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">c'</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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c20">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
;<br/>
<a NAME="E19:45_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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="c18">c</font> holds <br/><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c20">c'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c21">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:45_1_4"/><i><font color="Green" title="E50">A28</font></i>: 
<font color="Maroon" title="c21">k</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="c18">c</font>
 ;<br/>

<a NAME="E2:45_1_4"/><i><font color="Green" title="E51">A29</font></i>: 
<font color="Maroon" title="c21">k</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="c18">c</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:45_1_4"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">p''</font> = <font color="Maroon" title="c3">p'</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">c'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span> = 
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">p'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E14:45_1"><i><font color="Green" title="E48">A17</font></i></a>, <a class="txt" href="group_10.html#E2:45_1_4"><i><font color="Green" title="E51">A29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c22">p''</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p''</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c22">p''</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></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/>.= 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/>
<b>hence </b><a NAME="E5:45_1_4"/>
<font color="Maroon" title="c18">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c20">c'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">k</font>
 <b>by </b><i><a class="txt" href="group_10.html#E18:45_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="group_10.html#E1:45_1_4"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:45_1"/><b>then </b><i><font color="Green" title="E50">A30</font></i>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p'</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c20">c'</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E18:45_1"><i><font color="Green" title="E49">A27</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</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="c13">E</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Sum</a> <font color="Maroon" title="c20">c'</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#E12:45_1"><i><font color="Green" title="E47">A16</font></i></a>, <a class="txt" href="group_10.html#E20:45_1"><i><font color="Green" title="E50">A30</font></i></a>, <a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
<br/>.= 
 <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/>
<b>hence </b><a NAME="E22:45_1"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_10.html#E13:45"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="group_10.html#E35"><i><font color="Green" title="E31">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">E</font><b> such that </b><br/><a NAME="E19:45"/><i><font color="Green" title="E42">A31</font></i>: 
<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#K13" title="GROUP_10:func.13">the_orbit_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</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="txt" href="group_10.html#E17:45"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">HX</font> =  <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font>;<br/>
<a NAME="E20:45"/><i><font color="Green" title="E43">A32</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="c15">X</font>,<font color="Maroon" title="c14">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="c15">X</font>,<font color="Maroon" title="c14">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="E21:45"/>
 <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="c15">X</font>,<font color="Maroon" title="c14">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="group_2.html#T164" title="GROUP_2:th.164">GROUP_2:164</a></i>;<br/>
<a NAME="E22:45"/><b>then </b><i><font color="Green" title="E44">A33</font></i>: 
<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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</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="txt" href="group_10.html#E19:45"><i><font color="Green" title="E42">A31</font></i></a>, <a class="txt" href="group_10.html#E20:45"><i><font color="Green" title="E43">A32</font></i></a>, <a class="ref" href="group_2.html#D18" title="GROUP_2:def.18">GROUP_2:def 18</a></i>;<br/>
<a NAME="E23:45"/><i><font color="Green" title="E45">A34</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>;<br/>
<b>take </b>
 <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c17">H</font> =  <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font> as  <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> <b>by </b><i><a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E25:45"/><i><font color="Green" title="E46">A35</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</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/>
<a NAME="E26:45"/><i><font color="Green" title="E47">A36</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c18">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green" title="E48">A37</font></i>: 
 <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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">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/><b>thus </b><a NAME="E4:45_2"/>
contradiction
 <b>by </b><i><a class="txt" href="group_10.html#E22:45"><i><font color="Green" title="E44">A33</font></i></a>, <a class="txt" href="group_10.html#E3:45_2"><i><font color="Green" title="E48">A37</font></i></a>, <a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:45"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_10.html#E23:45"><i><font color="Green" title="E45">A34</font></i></a>, <a class="txt" href="group_10.html#E36"><i><font color="Green" title="E32">Lm6</font></i></a></i>;<br/>
<a NAME="E29:45"/><b>then </b><i><font color="Green" title="E48">A38</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E26:45"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E30:45"/><i><font color="Green" title="E49">A39</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">X</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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="c15">X</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_10.html#D10" target="_self" title="GROUP_10:def.10">Def10</a></i>;<br/>
<a NAME="E31:45"/>
<font color="Maroon" title="c15">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 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> :  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k1</font> </span>}</span> 
 
;<br/>
<a NAME="E32:45"/><b>then </b><i><font color="Green" title="E50">A40</font></i>: 
 ex <font color="Olive" title="b1">X'</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">X</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k1</font> )
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c18">X</font> = <font color="Maroon" title="c15">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_10.html#E32:45"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c18">X</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">x</font> = <font color="Maroon" title="c19">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_3"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H</font>
 ;<br/><b>then consider </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="c1">G</font><b> such that </b><br/><a NAME="E3:45_3"/><i><font color="Green" title="E51">A41</font></i>: 
( <font color="Maroon" title="c21">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="relset_1.html#K10" title="RELSET_1:func.10">.:</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> <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#E30:45"><i><font color="Green" title="E49">A39</font></i></a></i>;<br/><a NAME="E4:45_3"/><i><font color="Green" title="E52">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:45_3"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">T</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span>,<font color="Maroon" title="c18">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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">X</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E4:45_3"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/><a NAME="E6:45_3"/><b>then </b><i><font color="Green" title="E53">A43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">X</font>
 <b>by </b><i><a class="txt" href="group_10.html#E3:45_3"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">X1</font> = <font color="Maroon" title="c18">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">E</font> ;<br/><a NAME="E8:45_3"/>
<font color="Maroon" title="c14">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K4" title="GROUP_10:func.4">the_extension_of_left_translation_of</a> <font color="Maroon" title="c7">k1</font>,<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><a href="group_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E11:45"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="group_10.html#D5" target="_self" title="GROUP_10:def.5">Def5</a></i>;<br/><a NAME="E9:45_3"/><b>then </b><i><font color="Green" title="E54">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">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="c23">X1</font> <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_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c23">X1</font>
 <b>by </b><i><a class="txt" href="group_10.html#E11:45"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="group_10.html#D4" target="_self" title="GROUP_10:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">LO</font> =  <a href="group_10.html#K2" title="GROUP_10:func.2">the_left_operation_of</a> <font color="Maroon" title="c1">G</font> as   <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> ;<br/><b>reconsider </b><font color="Maroon" title="c25">g''</font> = <font color="Maroon" title="c22">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/><a NAME="E12:45_3"/><i><font color="Green" title="E55">A45</font></i>: 
<font color="Maroon" title="c24">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="group_10.html#NK3" title="GROUP_10:NK.3">the_left_translation_of</a> <font color="Maroon" title="c22">g</font>
 <b>by </b><i><a class="ref" href="group_10.html#D2" target="_self" title="GROUP_10:def.2">Def2</a></i>;<br/><a NAME="E13:45_3"/>
 ex <font color="Olive" title="b1">x'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span> &amp; <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> &amp; <font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">LO</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c26">x'</font> = <font color="Maroon" title="c20">x</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c26">x'</font>
;<br/>

<a NAME="E2:45_3_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:45_3_1"/>
( <font color="Maroon" title="c26">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span> &amp; <font color="Maroon" title="c26">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> )
 ;<br/>

<b>thus </b><a NAME="E4:45_3_1"/>
<font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">LO</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">x'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E12:45_3"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="topgrp_1.html#D1" title="TOPGRP_1:def.1">TOPGRP_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:45_3"/><b>then </b><i><font color="Green" title="E56">A46</font></i>: 
<font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <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="Maroon" title="c24">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c18">X</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>set </b><font color="Maroon" title="c26">h</font> = <font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x</font>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> = 
<font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c22">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c21">z</font>
<b>by </b><i><a class="txt" href="group_10.html#E3:45_3"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E16:45_3"/>
<font color="Maroon" title="c21">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">X</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E6:45_3"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="group_10.html#E9:45_3"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="group_10.html#E14:45_3"><i><font color="Green" title="E56">A46</font></i></a>, <a class="ref" href="group_2.html#T34" title="GROUP_2:th.34">GROUP_2:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E37:45"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">X</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E38:45"/><b>then </b>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c18">X</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>
<a NAME="E39:45"/><b>then </b><i><font color="Green" title="E51">A47</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">H</font> <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="c18">X</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E37"><i><font color="Green" title="E33">Lm7</font></i></a></i>;<br/>
<a NAME="E40:45"/><i><font color="Green" title="E52">A48</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E32:45"><i><font color="Green" title="E50">A40</font></i></a>, <a class="txt" href="group_10.html#E39:45"><i><font color="Green" title="E51">A47</font></i></a>, <a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E41:45"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c17">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E29:45"><i><font color="Green" title="E48">A38</font></i></a>, <a class="txt" href="group_10.html#E40:45"><i><font color="Green" title="E52">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E42:45"/><b>then </b>
<font color="Maroon" title="c17">H</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></i>;<br/>
<a NAME="E43:45"/><b>then </b><i><font color="Green" title="E53">A49</font></i>: 
 <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font> <a href="group_10.html#R4" title="GROUP_10:pred.4">is_p-group_of_prime</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
 <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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span> = 
<span class="p1">(<span class="default"><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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default">1 <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> <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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E25:45"><i><font color="Green" title="E46">A35</font></i></a>, <a class="ref" href="xcmplx_1.html#T108" title="XCMPLX_1:th.108">XCMPLX_1:108</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <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> <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="c15">X</font>,<font color="Maroon" title="c14">T</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_10.html#E29:45"><i><font color="Green" title="E48">A38</font></i></a>, <a class="txt" href="group_10.html#E40:45"><i><font color="Green" title="E52">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
<br/>.= 
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p'</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
<br/>.= 
<font color="Maroon" title="c4">m</font>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T89" title="XCMPLX_1:th.89">XCMPLX_1:89</a></i>
;<br/>
<b>hence </b><a NAME="E45:45"/>
 <a href="group_10.html#K11" title="GROUP_10:func.11">the_strict_stabilizer_of</a> <font color="Maroon" title="c15">X</font>,<font color="Maroon" title="c14">T</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:45"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="group_10.html#E43:45"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>


</div>
