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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font>, <font color="Olive" title="b2">M</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">M</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Olive" title="b1">N</font> holds <br/><font color="Olive" title="b1">N</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">N</font>,<font color="Olive" title="b2">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c3">M</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <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"> ( <font color="Maroon" title="c3">M</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">N</font> implies <font color="Maroon" title="c2">N</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">M</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">N</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="c2">N</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span><br/>

<a NAME="E2:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D1" target="_self" title="GROUP_6:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">M</font> as   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">J</font> = <font color="Maroon" title="c2">N</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="group_6.html#T34" target="_self" title="GROUP_6:th.34">Th34</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">S</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span>; <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="c6">S</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">S</font></span><br/><span class="kw">thus </span><a NAME="E1:42_1"/>
<font color="Maroon" title="c6">S</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">J</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">S</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">J</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c5">J</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="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">S</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">T</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:42_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">S</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">T</font>
 <span class="kw">and </span><br/><a NAME="E4:42_1_1"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c8">T</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">J</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E6:42_1_1"/>
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">I</font>
 <span class="kw">and </span><br/><a NAME="E7:42_1_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
<font color="Maroon" title="c8">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E4:42_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="group_6.html#T28" target="_self" title="GROUP_6:th.28">Th28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c9">c</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">consider </span><font color="Maroon" title="c11">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E10:42_1_1"/>
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E11:42_1_1"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c6">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">e</font> = <font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span>;<br/>
<a NAME="E12:42_1_1"/>
( <font color="Maroon" title="c9">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T4" target="_self" title="GROUP_6:th.4">Th4</a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E13:42_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_5.html#T3" title="GROUP_5:th.3">GROUP_5:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<a NAME="E15:42_1_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">V</font> = <font color="Maroon" title="c4">I</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c13">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">J</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="group_6.html#T16" target="_self" title="GROUP_6:th.16">Th16</a></span>;<br/>
<a NAME="E17:42_1_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c14">V</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">J</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">V</font> = <font color="Maroon" title="c14">V</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font></span>)</span> <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/>
<a NAME="E19:42_1_1"/>
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">I</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c9">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T2" target="_self" title="GROUP_6:th.2">Th2</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c7">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E11:42_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E7:42_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c10">d</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T129" title="GROUP_2:th.129">GROUP_2:129</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T12" title="GROUP_3:th.12">GROUP_3:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">M</font></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T15" title="GROUP_3:th.15">GROUP_3:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T118" title="GROUP_2:th.118">GROUP_2:118</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T129" title="GROUP_2:th.129">GROUP_2:129</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T129" title="GROUP_2:th.129">GROUP_2:129</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">M</font></span>)</span> <a href="group_2.html#K5" title="GROUP_2:func.5">*</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T140" title="GROUP_3:th.140">GROUP_3:140</a></span>
<br/>.= 
<font color="Maroon" title="c3">M</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T128" title="GROUP_2:th.128">GROUP_2:128</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c11">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T13" title="GROUP_3:th.13">GROUP_3:13</a></span>
<br/>.= 
<font color="Maroon" title="c15">V</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E15:42_1_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="group_6.html#D4" target="_self" title="GROUP_6:def.4">Def4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E21:42_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">J</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42_1_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</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>
<span class="kw">hence </span><a NAME="E6:42"/>
<font color="Maroon" title="c2">N</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c3">M</font> <a href="group_6.html#K1" title="GROUP_6:func.1">`*`</a> </span>)</span> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_3.html#T141" title="GROUP_3:th.141">GROUP_3:141</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>
