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<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 ;<br/>

<b>thus </b><a NAME="E2:6"/>
( ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">p</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">p</font> ) implies <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E7">A2</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">p</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">p</font>
 ;<br/>

<a NAME="E2:6_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">p</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> <b> such that </b><br/><a NAME="E2:6_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:6_1"><i><font color="Green" title="E7">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c4">p</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">p1</font>
;<br/>

<b>thus </b><a NAME="E4:6_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">p1</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:6_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1"/>
<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T11" target="_self" title="GR_CY_2:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:6"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>consider </b><font color="Maroon" title="c4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E5:6"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:6"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="gr_cy_2.html#T11" target="_self" title="GR_CY_2:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">n1</font> =  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E7:6"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c5">n1</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="txt" href="gr_cy_2.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E8:6"/>
<font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</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="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<i><font color="Green" title="E10">A7</font></i>: <font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E5:6"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:6"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c5">n1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<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></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></i>
<br/>.= 
<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> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:6"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
<br/>.= 
<font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c6">p</font>
;<br/>

<b>thus </b><a NAME="E11:6"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:6"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/>


</div>
