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

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="gr_cy_3.html#V1" title="GR_CY_3:attr.1">Safe</a>  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <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">q</font> being  <a href="gr_cy_3.html#V2" title="GR_CY_3:attr.2">Sophie_Germain</a>  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> ex <font color="Olive" title="b2">H1</font>, <font color="Olive" title="b3">H2</font>, <font color="Olive" title="b4">Hq</font>, <font color="Olive" title="b5">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b3">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b4">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b5">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">q</font> &amp; ( for <font color="Olive" title="b6">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b6">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b2">H1</font> or <font color="Olive" title="b6">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b3">H2</font> or <font color="Olive" title="b6">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b4">Hq</font> or <font color="Olive" title="b6">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b5">H2q</font> ) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">q</font> being   <a href="gr_cy_3.html#V2" title="GR_CY_3:attr.2">Sophie_Germain</a>  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E2:20"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_3.html#T13" target="_self" title="GR_CY_3:th.13">Th13</a></span>;<br/>
<a NAME="E3:20"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</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="int_7.html#T31" title="INT_7:th.31">INT_7:31</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E6:20"/>
 <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="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="gr_cy_3.html#T14" target="_self" title="GR_CY_3:th.14">Th14</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:20"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">and </span><br/><a NAME="E9:20"/>
 <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="c4">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is   <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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="gr_cy_3.html#T14" target="_self" title="GR_CY_3:th.14">Th14</a></span>;<br/>
<a NAME="E10:20"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></span>;<br/>
<a NAME="E11:20"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <span class="p2">(<span class="default"><a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T81" title="GROUP_2:th.81">GROUP_2:81</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">H1</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font><span class="kw"> such that </span><br/><a NAME="E13:20"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>
<a NAME="E14:20"/>
 <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> is   <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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T63" title="GROUP_2:th.63">GROUP_2:63</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font><span class="kw"> such that </span><br/><a NAME="E16:20"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c6">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a></span>;<br/>
<a NAME="E17:20"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <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">then </span><span class="kw">consider </span><font color="Maroon" title="c7">H2</font>, <font color="Maroon" title="c8">Hq</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font><span class="kw"> such that </span><br/><a NAME="E19:20"/><span class="lab"><font color="Green" title="E20">A8</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">and </span><br/><a NAME="E20:20"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c8">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E17">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">q</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">H1</font>, <font color="Olive" title="b2">H2</font>, <font color="Olive" title="b3">Hq</font>, <font color="Olive" title="b4">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b2">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b3">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b4">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b5">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b5">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b1">H1</font> or <font color="Olive" title="b5">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b2">H2</font> or <font color="Olive" title="b5">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b3">Hq</font> or <font color="Olive" title="b5">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b4">H2q</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">H1</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">H2</font>, <font color="Olive" title="b2">Hq</font>, <font color="Olive" title="b3">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b2">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b3">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b4">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b4">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Olive" title="b4">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b1">H2</font> or <font color="Olive" title="b4">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b2">Hq</font> or <font color="Olive" title="b4">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b3">H2q</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">H2</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">Hq</font>, <font color="Olive" title="b2">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b2">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b3">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b3">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Olive" title="b3">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Olive" title="b3">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b1">Hq</font> or <font color="Olive" title="b3">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b2">H2q</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c8">Hq</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">H2q</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font> st <br/>(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c8">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b2">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Olive" title="b2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Olive" title="b2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Olive" title="b2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Olive" title="b1">H2q</font> ) ) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">H2q</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#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c8">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c6">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b1">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> ) ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">H</font> be   <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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>; <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="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c10">G1</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font><span class="kw"> such that </span><br/><a NAME="E2:20_1"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c10">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">and </span><br/><a NAME="E3:20_1"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</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>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</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="c10">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="group_2.html#T178" title="GROUP_2:th.178">GROUP_2:178</a>, <a class="ref" href="gr_cy_2.html#T28" title="GR_CY_2:th.28">GR_CY_2:28</a></span>;<br/><a NAME="E4:20_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c10">G1</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> or  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="gr_cy_3.html#T1" target="_self" title="GR_CY_3:th.1">Th1</a>, <a class="ref" href="group_2.html#T178" title="GROUP_2:th.178">GROUP_2:178</a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_1"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</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"> ( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_1"/>
( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E23">A11</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_2"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_2"/>
( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E23">A11</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_3"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 ; <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="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_3"/>
( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E23">A11</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1_4"/>
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font>
 ; <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="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_4"/>
( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E3:20_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E4:20_1"><span class="lab"><font color="Green" title="E23">A11</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:20_1"/>
( <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Maroon" title="c9">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> )
 ; <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>
<span class="kw">hence </span><a NAME="E22:20"/>
(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c5">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c8">Hq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c6">H2q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c2">q</font> &amp; ( for <font color="Olive" title="b1">H</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  <a href="int_7.html#K3" title="INT_7:func.3">Z/Z*</a> <font color="Maroon" title="c1">p</font>  holds <br/>( <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c5">H1</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c7">H2</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c8">Hq</font> or <font color="Olive" title="b1">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c6">H2q</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:20"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E19:20"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E20:20"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="txt" href="#E13:20"><span class="lab"><font color="Green" title="E18">A6</font></span></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/>


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