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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b3">l</font> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">l</font></span><br/><b>let </b><font color="Maroon" title="c2">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font>, <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">l</font> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">l</font></span><br/><b>let </b><font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">l</font> implies  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> )</span><br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E40">A1</font></i>: 
( <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> &amp;  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">H</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="c2">z</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_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<b>set </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c3">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">l</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c5">H</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_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E41">A2</font></i>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>
<a NAME="E5:48"/>
<font color="Maroon" title="c2">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</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">z</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#T8" title="GR_CY_2:th.8">GR_CY_2:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c2">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">H</font> <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:48"/>
 <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">z</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>  <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="c8">z1</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#T9" title="GR_CY_2:th.9">GR_CY_2:9</a></i>;<br/>
<a NAME="E8:48"/><b>then </b>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font>
 
<b>by </b><i><a class="txt" href="#E4:48"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="gr_cy_2.html#T14" title="GR_CY_2:th.14">GR_CY_2:14</a></i>;<br/>
<b>hence </b><a NAME="E9:48"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="txt" href="#E1:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="group_8.html#T3" title="GROUP_8:th.3">GROUP_8:3</a>, <a class="ref" href="group_8.html#T5" title="GROUP_8:th.5">GROUP_8:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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