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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <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>   st <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> holds <br/> ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> implies  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">p9</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:47"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T12" title="NAT_4:th.12">NAT_4:12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">P</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c4">P</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#T12" target="_self" title="GROUP_10:th.12">Th12</a></span>;<br/>
<a NAME="E5:47"/>
<font color="Maroon" title="c4">P</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">H</font> being   <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><span class="kw"> such that </span><br/><a NAME="E7:47"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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> <span class="kw"> such that </span><br/><a NAME="E10:47"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">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="c6">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="group_10.html#D17" target="_self" title="GROUP_10:def.17">Def17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:47"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c6">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:47_1"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>;<br/><a NAME="E3:47_1"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></span>;<br/><span class="kw">hence </span><a NAME="E4:47_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E13:47"/><span class="kw">then </span>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E14:47"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:47"/><span class="kw">then </span>
1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">r9</font> = <font color="Maroon" title="c6">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/>
<a NAME="E17:47"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p9</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c7">r9</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E18:47"/><span class="kw">then </span>
1 <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="c7">r9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E19:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A7</font></span>: 
1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">r9</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">H9</font> =  <a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font>;<br/>
<span class="lab"><font color="Green" title="E43">A8</font></span>:  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T177" title="GROUP_2:th.177">GROUP_2:177</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p2">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T180" title="GROUP_2:th.180">GROUP_2:180</a></span>
;<br/>
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c6">r</font> = 
<font color="Maroon" title="c2">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<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="c7">r9</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></span>
;<br/>
<a NAME="E22:47"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E19:47"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E24:47"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="group_2.html#K7" title="GROUP_2:func.7">(Omega).</a> <font color="Maroon" title="c5">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#T31" title="GR_CY_1:th.31">GR_CY_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">H99</font> =  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">g</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<a NAME="E26:47"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c10">H99</font> is   <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_2.html#T10" title="GR_CY_2:th.10">GR_CY_2:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">H99</font> = <font color="Maroon" title="c10">H99</font> as   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">n</font> =  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">H99</font>;<br/>
<div><span class="lab"><font color="Green" title="E46">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:47_4"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">H99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:47_4"/><span class="kw">then </span>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E3:47_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:47"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="group_1.html#T85" title="GROUP_1:th.85">GROUP_1:85</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E29:47"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">H99</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></span>;<br/>
<a NAME="E30:47"/><span class="kw">then </span>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">H99</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:47"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E31:47"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">H99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E20:47"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="group_10.html#E39"><span class="lab"><font color="Green" title="E35">Lm8</font></span></a>, <a class="ref" href="group_2.html#T178" title="GROUP_2:th.178">GROUP_2:178</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">H999</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c11">H99</font><span class="kw"> such that </span><br/><a NAME="E33:47"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c13">H999</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p9</font>
 <span class="kw">and </span><br/><a NAME="E34:47"/>
for <font color="Olive" title="b1">G2</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c11">H99</font>  st  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p9</font> holds <br/><font color="Olive" title="b1">G2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c13">H999</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:47"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="gr_cy_2.html#T28" title="GR_CY_2:th.28">GR_CY_2:28</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">h9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">H999</font><span class="kw"> such that </span><br/><a NAME="E36:47"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c14">h9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:47"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="group_8.html#T1" title="GROUP_8:th.1">GROUP_8:1</a></span>;<br/>
<a NAME="E37:47"/>
<font color="Maroon" title="c11">H99</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="group_2.html#T65" title="GROUP_2:th.65">GROUP_2:65</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">H999</font> = <font color="Maroon" title="c13">H999</font> as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T65" title="GROUP_2:th.65">GROUP_2:65</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">h9</font> = <font color="Maroon" title="c14">h9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">H999</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">h</font> = <font color="Maroon" title="c16">h9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_1.html#K6" title="GROUP_1:func.6">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></span><br/>

<span class="kw">thus </span><a NAME="E41:47"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:47"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="group_8.html#T5" title="GROUP_8:th.5">GROUP_8:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
